observation and estimation study for sensorless control of
TRANSCRIPT
University of LouisvilleThinkIR: The University of Louisville's Institutional Repository
Electronic Theses and Dissertations
8-2017
Observation and estimation study for sensorlesscontrol of linear vapor compressors.Joseph LathamUniversity of Louisville
Follow this and additional works at: https://ir.library.louisville.edu/etd
Part of the Controls and Control Theory Commons
This Doctoral Dissertation is brought to you for free and open access by ThinkIR: The University of Louisville's Institutional Repository. It has beenaccepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of ThinkIR: The University of Louisville's InstitutionalRepository. This title appears here courtesy of the author, who has retained all other copyrights. For more information, please [email protected].
Recommended CitationLatham, Joseph, "Observation and estimation study for sensorless control of linear vapor compressors." (2017). Electronic Theses andDissertations. Paper 2745.https://doi.org/10.18297/etd/2745
OBSERVATION AND ESTIMATION STUDY FOR SENSORLESS
CONTROL OF LINEAR VAPOR COMPRESSORS
By
Joseph Latham
B.S., Western Kentucky University, 2012
M.S., University of Louisville, 2014
A Dissertation
Submitted to the Faculty of the
J.B. Speed School of Engineering of the University of Louisville
in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
in Electrical Engineering
Department of Electrical and Computer Engineering
University of Louisville
Louisville, Kentucky
August 2017
ii
OBSERVATION AND ESTIMATION STUDY FOR SENSORLESS
CONTROL OF LINEAR VAPOR COMPRESSORS
By
Joseph Latham
B.S., Western Kentucky University, 2012
M.S., University of Louisville, 2014
A Dissertation Approved on
July 14, 2017
by the following Dissertation Committee:
__________________________________
Michael McIntyre, Dissertation Director
__________________________________
John Naber
__________________________________
Tamer Inanc
__________________________________
Chris Richards
iii
ABSTRACT
OBSERVATION AND ESTIMATION STUDY FOR SENSORLESS CONTROL OF LINEAR VAPOR
COMPRESSORS
Joseph Latham
July 14, 2017
Linear vapor compressors have become widely investigated for refrigeration applications due to
their high efficiency in comparison to the more common rotary type compressors. However, the nature of
the linear compressor adds complexity to the control of these machines. The unconstrained motion of the
piston in a linear compressor allows for continuous modulation of the compressor output, but requires
knowledge of the mechanical dynamics to effectively control the compressor and prevent collision of the
piston with the cylinder head. This control is made more difficult by the highly nonlinear nature of the
force of gas compression acting against the piston. As this gas force changes so does the resonant
frequency of the system. Efficient control of the compressor requires knowledge and tracking of this
resonant frequency in addition to other objectives. Sensorless control of the system is preferred for
reliability, ease of production, and cost effectiveness. To this end a series of nonlinear observers and a
combination of controllers have been developed for the linear vapor compressor.
iv
TABLE OF CONTENTS
PAGE
ABSTRACT ......................................................................................................................................... iii
LIST OF FIGURES ............................................................................................................................. vii
1 INTRODUCTION ......................................................................................................................... 1
2 LINEAR VAPOR COMPRESSOR SYSTEM MODEL ............................................................... 5
2.1 THERMODYNAMIC SYSTEM MODEL ........................................................................... 6
2.2 MECHANICAL SYSTEM MODEL .................................................................................... 7
2.3 ELECTRICAL SYSTEM MODEL ...................................................................................... 8
3 SIMULATION PLATFORM ...................................................................................................... 11
4 EXPERIMENTAL PLATFORM ................................................................................................. 13
4.1 HARDWARE DESIGN ...................................................................................................... 13
4.2 SOFTWARE DESIGN........................................................................................................ 14
5 PARAMETER ESTIMATION .................................................................................................... 16
5.1 ELECTRICAL PARAMETERS ......................................................................................... 16
5.1.1 ESTIMATOR DESIGN .................................................................................................. 16
5.1.2 EXPERIMENTAL RESULTS ....................................................................................... 19
5.2 MECHANICAL PARAMETERS ....................................................................................... 21
5.2.1 ESTIMATOR DESIGN .................................................................................................. 21
5.2.2 SIMULATION RESULTS ............................................................................................. 24
v
6 REAL-TIME OBSERVERS ........................................................................................................ 26
6.1 VELOCITY OBSERVER ................................................................................................... 26
6.1.1 OBSERVER DEVELOPMENT ..................................................................................... 26
6.1.2 STABILITY ANALYSIS ............................................................................................... 28
6.1.3 EXPERIMENTAL RESULTS ....................................................................................... 28
6.2 ACCELERATION OBSERVER ........................................................................................ 29
6.2.1 OBSERVER DEVELOPMENT ..................................................................................... 30
6.2.2 STABILITY ANALYSIS ............................................................................................... 31
6.2.3 EXPERIMENTAL RESULTS ....................................................................................... 32
6.3 TWO-STAGE OBSERVER ................................................................................................ 33
6.3.1 OBSERVER DEVELOPMENT ..................................................................................... 33
6.3.2 EXPERIMENTAL RESULTS ....................................................................................... 33
6.4 GAS FORCE OBSERVER ................................................................................................. 34
6.4.1 OBSERVER DEVELOPMENT ..................................................................................... 35
6.4.2 STABILITY ANALYSIS ............................................................................................... 36
6.4.3 EXPERIMENTAL RESULTS ....................................................................................... 37
6.5 POSITION AND PRESSURE OBSERVER ...................................................................... 38
6.5.1 RELATIVE POSITION OBSERVER ............................................................................ 38
6.5.2 LINEARLY PARAMETERIZED GAS FORCE ........................................................... 39
6.5.3 OBSERVER DEVELOPMENT ..................................................................................... 40
6.5.4 STABILITY ANALYSIS ............................................................................................... 42
6.5.5 EXPERIMENTAL RESULTS ....................................................................................... 44
6.5.6 INCOMPLETE CYCLE MODIFICATION ................................................................... 48
vi
6.5.7 SOFT CRASH MODIFICATION .................................................................................. 49
7 PER-CYCLE OBSERVATIONS ................................................................................................ 52
7.1 PHASE OBSERVATION ................................................................................................... 52
7.2 STROKE LENGTH OBSERVATION ............................................................................... 53
7.3 BOTTOM DEAD CENTER OBSERVATION .................................................................. 53
7.4 TOP DEAD CENTER OBSERVATION............................................................................ 54
8 REGULATION CONTROLLERS .............................................................................................. 55
8.1 CURRENT CONTROLLER ............................................................................................... 56
8.2 RESONANCE CONTROLLER ......................................................................................... 56
8.3 TOP DEAD CENTER CONTROLLER ............................................................................. 57
8.4 CONTROLLER PRIORITY ............................................................................................... 58
8.5 EXPERIMENTAL RESULTS ............................................................................................ 59
9 TRACKING CONTROLLERS ................................................................................................... 63
9.1 CURRENT CONTROLLER ............................................................................................... 63
9.1.1 CONTROL DEVELOPMENT ....................................................................................... 63
9.1.2 STABILITY ANALYSIS ............................................................................................... 65
9.1.3 SIMULATION RESULTS ............................................................................................. 65
CONCLUSION..................................................................................................................................... 68
REFERENCES ..................................................................................................................................... 69
APPENDIX .......................................................................................................................................... 71
CURRICULUM VITAE ....................................................................................................................... 74
vii
LIST OF FIGURES
Figure 1. Representation of a typical linear vapor compressor [16]. ..................................................... 5
Figure 2. Pressure-displacement curve for a linear vapor compressor [16]. ........................................... 6
Figure 3. Equivalent electrical circuit model of a linear motor. ............................................................. 9
Figure 4. H-bridge inverter topology. ..................................................................................................... 9
Figure 5. Block diagram of experimental hardware setup. .................................................................. 13
Figure 6. Diagram of software architecture and hardware integration for NI CompactRIO platform. 15
Figure 7. Estimation results for the electrical parameters πΌ, π π, πΏπ. ..................................................... 20
Figure 8. Simulation results for adaptive least-squares mechanical parameter estimation scheme. .... 25
Figure 9. Convergence of observer error πππ‘. ..................................................................................... 29
Figure 10. Comparison of velocity, measured π₯(π‘) and observed π₯π(π‘). ........................................... 29
Figure 11. Comparison of velocity, measured π₯(π‘) and observed π₯π(π‘). ........................................... 32
Figure 12. Comparison acceleration, measured π₯(π‘) and observed π₯π(π‘). ......................................... 32
Figure 13. Uncertainty observer ππ(π‘). .............................................................................................. 33
Figure 14. Velocity observers π₯π(π‘) and π₯π(π‘). ................................................................................. 34
Figure 15. Comparison of acceleration, measured π₯(π‘) and observed π₯π(π‘). ..................................... 34
Figure 16. Comparison of actual velocity π₯(π‘) and observed velocity π₯πΉ(π‘). .................................... 37
Figure 17. Observed gas force πΉππ‘. ..................................................................................................... 37
Figure 18. Comparison of relative position signal π₯π‘ and observer position signal π₯π‘ during initial
transient. ........................................................................................................................................................ 45
Figure 19. Comparison of velocity signal π₯π‘ and observer velocity π₯π‘ during initial transient. ........... 45
Figure 20. Convergence of observer error signals π₯π‘, π₯π‘, ππ‘ during initial transient. ........................... 46
Figure 21. Convergence of parameter estimate π₯ππ·πΆ to actual top dead center value π₯ππ·πΆ. ............ 46
Figure 22. Convergence of parameter estimates ππ , ππ to actual pressures ππ , ππ. ........................... 46
viii
Figure 23. Comparison of position observer including top dead center estimate (π₯π‘ + π₯ππ·πΆ) with
actual absolute position π₯(π‘) under observer steady-state. ............................................................................ 47
Figure 24. Instantaneous chamber pressure observer ππ‘ during observer steady-state, shown with
estimated suction and discharge pressures ππ , ππ. ........................................................................................ 47
Figure 25. Instantaneous gas force observer πΉππ‘ during observer steady-state as calculated from
estimated pressures ππ , ππ. ........................................................................................................................... 47
Figure 26. Example of soft crash condition for linear compressor. ...................................................... 49
Figure 27. Block diagram of system level sensorless control scheme. ................................................. 55
Figure 28. Performance of proportional-integral peak current controller with varying setpoint
determined by top dead center controller. ..................................................................................................... 61
Figure 29. Amplitude π0π‘ of the voltage excitation π£ππ‘ calculated by the current controller............. 61
Figure 30. Measured and observed phase difference between the piston velocity and measured current
with corresponding desired phase. ................................................................................................................ 62
Figure 31. Frequency π0π‘ of the voltage excitation set by the resonance controller as it attempts to
achieve the desired phase shown in Fig. 21. .................................................................................................. 62
Figure 32. Measured and observed top dead center with corresponding desired top dead center value.
....................................................................................................................................................................... 62
Figure 33. Convergence of actual current πΌπ‘ to the current trajectory πΌππ‘. ........................................... 66
Figure 34. Convergence of current error ππΌ(π‘) to zero. ........................................................................ 66
Figure 35. Controller voltage command π£π(π‘) and corresponding H-bridge inverter output. .............. 67
Figure 36. Comparison of observed velocity π₯πΌ(π‘) with actual velocity π₯(π‘). ..................................... 67
1
1 INTRODUCTION
Rising energy costs and environmentally motivated legislation has led to increased research and
development in the area of high efficiency domestic appliances. In the realm of refrigeration this has
resulted in a renewed interest in linear vapor compressors over the historically more common rotary type.
Linear compressors offer a gain in efficiency over the rotary compressor due to the fact that the motor force
is inherently linear and therefore requires no mechanical conversion to actuate the piston [1]. Along with
other factors this has allowed linear compressors to demonstrate improvements in efficiency of 10-20%
over that of traditional rotary compressors driven by induction motors [2].
Despite these gains in efficiency there is an increase in complexity in the use of a linear compressor
over a rotary device. Whereas in a rotary compressor the piston stroke (peak-to-peak displacement) is
defined and constrained by the diameter of rotation, in a linear compressor the piston is free, allowing the
stroke to be variable. This is an advantage in that it allows for the modulation of the compressor output,
however it adds complexity in that the force of gas compression is free to change the pistonβs path. These
effects necessitate an additional level of control to achieve efficient and stable operation of a linear
compressor [3]. This control necessitates knowledge of the piston dynamics, which could most directly be
obtained through use of some type of sensor as in [4], [5]. However, due to the extreme environment in a
refrigerator compressor, i.e. refrigerant and oil, it is not cost effective to place a sensor inside the
compressor. Moreover, the shell is hermetically sealed and if a sensor is to be used, at least two additional
wires are required to penetrate the shell, which may not be acceptable in production. Hence sensorless
control of the piston is generally preferred [6].
Efforts in this space include an external self-sensor circuit which utilizes voltage and current
measurements to create a position signal [7] as well as algorithms such as those presented in [6], [8], [9],
[10] which utilize the same signals to arrive at a position signal computationally. However, each of these
methods has one or more significant weaknesses. For instance, [6] requires the taking of numerical
2
derivatives which amplifies the noise present in measured signals. The methods in [6], [8], and [9] require
integration of signals which, due to unavoidable biases in voltage detection circuits, may become
unbounded. In [10] this problem is addressed and is avoided by utilizing a high-pass filter to eliminate any
DC bias in the velocity signal being integrated. However, since the real position signal has a significant
DC component which varies with the force of gas compression, this method is incapable of accurately
estimating this component and thus the absolute position.
Control of a linear compressor typically consists of two main objectives; resonance control to ensure
maximum efficiency and stroke control to prevent collision and modulate output. Resonance control
requires identification and tracking of the system resonant frequency which changes with gas force. In [11]
this tracking is achieved using a perturb and observe algorithm to search for the maximum power point.
The nature of this algorithm results in βhuntingβ or undesirable oscillation in the frequency. In [12] the
resonant frequency is calculated directly from the system mechanical dynamics, but requires a linearization
of the gas force, which is not accurate, or at least very limited given that gas force is highly nonlinear. A
method was developed for identifying resonance via the relative phase between motor current and piston
position in [13] and similarly between current and velocity in [14], [15]. Both of these methods require
signals which must be estimated if the methods are to be implemented sensorlessly. A current controller has
been designed using a hybrid proportional-integral/neural network controller a sinusoidal current trajectory
once the resonant frequency has been identified [16].
The second objective is control of the top dead center position, which corresponds to the minimum
distance between the piston and the cylinder head. The purpose of this control is two-fold, modulation of
the compressor output and avoidance of collision with the cylinder head. For the purpose of collision
avoidance especially, knowledge of the absolute position is critical, which makes the majority of existing
methods presented unsuitable.
In this work a series of nonlinear observers are utilized to obtain observations for signals in the
mechanical dynamics. Each of these observers includes a robust integral of sign of error (RISE) term to
compensate for uncertainties in the system dynamics. The first observer uses a motor current measurement
along with knowledge of the motor voltage and the systemβs electrical model to arrive at an observed
velocity. Unlike other methods, which essentially solve the electrical dynamic equation for velocity, this is
3
a robust feedback based observer. The second observer utilizes a velocity measurement along with
knowledge of the systemβs mechanical model to arrive at an observation for piston acceleration. This
observer avoids the need to take noisy numerical derivatives to obtain an acceleration signal such as in [6].
A Lyapunov stability analysis is presented in [17] to prove stability and convergence for these two
observers with experimental results providing further validation. By using the observed velocity from the
first observer in place of the velocity measurement required by the second observer the two observers are
able to run in parallel using only a single current measurement. A gas force observer, derived separately,
requires measurement of both position and velocity and so cannot be implemented sensorlessly either alone
or in parallel. As such this observer is intended for laboratory use where full state measurements are
available and is useful for characterization of the gas force.
The developed velocity and acceleration observers are then utilized to obtain observations of per-
cycle system states which are relevant to the system control objectives. The observed velocity signal is used
to obtain knowledge of the phase difference between the piston velocity with respect to the motor current.
This observed phase difference is used as an indicator of system resonance and is used to achieve the
system level control objective of maximum efficiency via a search algorithm based controller.
The observed acceleration signal is used in tandem with the observed velocity and motor current to
obtain knowledge of the minimum position of the piston, known as top dead center. Unlike the majority of
methods present in the literature this is an absolute position estimation which includes the DC bias induced
by the force of gas compression and requires no numerical derivatives. Thus this observed top dead center
position can be used to achieve the top dead center control objective and thus prevent collision and
modulate compressor output. This controller uses the peak motor current as its control input since this
current is proportional to the motor force which actuates the motor. This desired motor current is then used
as a reference to a third controller which achieves regulation control of the amplitude of the motor current.
Alternatively, a tracking controller is also presented which achieves control of the current waveform via a
trajectory whose frequency is set by the resonance controller. By controlling the current over the entire
cycle instead of only at its peak this controller offers a more sophisticated and direct control over the piston
dynamics at the cost of added complexity.
4
As the afore-mentioned observers require a priori knowledge of the system model and parameters, this
work also includes an investigation of estimation schemes for the parameters of the various dynamics
which comprise the linear compressor system model. Least-squares algorithms are commonly used to
calculate parameters over a specified interval of samples. In [17] we see how this method can be applied to
time-varying parameters in linear systems and [8] shows an implementation specifically for linear
compressors.
For this work an adaptive least-squares algorithm as described in [18] was investigated. This
algorithm has been shown to be effective in estimation problems for other systems with similar models
[19], [20]. Due to the fact that gas force is highly nonlinear and difficult to measure, this method is
considered only for the electrical dynamics, requiring measurement of the motor current, voltage, and
piston velocity. The need for a velocity measurement makes this method unsuitable for sensorless
application, but it may be usable for production testing. It should be noted that due its dependence on
parameter values it is not possible to use the velocity observer developed in place of this measurement.
Doing so would create a null-space [21] in which a unique solution is unattainable for either problem.
The remainder of this work will proceed as follows: in Section 2 the system model will be presented
with its constituent thermodynamic, mechanical, and electrical dynamic equations, in Sections 3 and 4 the
simulation and experimental platforms, respectively, which were used in testing of the developed
algorithms will be detailed; in Section 5 parameter estimation schemes for the electrical and mechanical
parameters of the compressor are developed using the adaptive least-squares algorithm; in Section 6 real-
time observers are developed for the piston velocity, acceleration, force of gas compression, position and
pressures; in Section 7 we show how the velocity and acceleration observers can be used to obtain
observations of relevant per-cycle system states; in Section 8 we show how these per-cycle observations
can be utilized to develop regulation controllers which together achieve the joint system level of objectives
of stroke control and resonance tracking; in Section 9 a more sophisticated current controller is presented as
a possible replacement for the regulation controller put forth in Section 8. Concluding remarks are
provided following this section.
5
2 LINEAR VAPOR COMPRESSOR SYSTEM MODEL
A representation of the construction of a linear compressor is given in Figure 1. As shown, this device
consists of a compression chamber whose piston is actuated by a linear motor. This construction creates
three levels of coupled dynamics within the system. At the lowest level the oscillation of the piston in the
chamber causes compression, decompression, suction, and discharge of the working gas as it cyclically
pumps gas from lower to higher pressures. The thermodynamics which dictate this compression cycle
cause a resultant force on the piston which couple into its mechanical dynamics. This mechanical dynamic
is then cross-coupled with the electrical dynamics of the current induced in the stator coils via the motor
force coupled into the mechanical dynamics and the back EMF coupled into the electrical dynamics.
Figure 1. Representation of a typical linear vapor compressor [16].
6
2.1 THERMODYNAMIC SYSTEM MODEL
The thermodynamics which dictate the chamber pressure π(π‘) can be represented by the pressure-
displacement diagram shown in Figure 2. This diagram shows the chamber pressure as a function of the
piston position π₯(π‘), as it relates to the discharge and suction pressures, ππ β β and ππ β β, respectively.
We can see from this figure that the chamber pressure π(π‘) is always greater than or equal to the suction
pressure ππ .
Figure 2. Pressure-displacement curve for a linear vapor compressor [16].
As shown in Figure 2 four different stages make up the compression cycle, resulting in a piecewise
continuous chamber pressure π(π‘). The resultant force imposed on the piston by this gas compression,
denoted as πΉπ(π‘) β β, is related to the pressure difference across the piston head as shown in the following
equation:
πΉπ β π΄π(π(π‘) β ππ) (1)
where π(π‘) β β is the pressure of the compression chamber, π΄π β β is the cross-sectional area of the
piston, and ππ β β is the pressure on the back side of the piston head. In a typical compression cycle, ππ
can be assumed to be equal to the suction pressure ππ . From the assumption that ππ = ππ , we can see from
Fig. 2 and (1) that πΉπ β₯ 0 for all π‘. A detailed description of the piecewise chamber pressure and gas force
7
through the four stages shown in Figure 2 is given in Table 1 below. Note that the parameter π β β
denotes the specific heat ratio for the working gas.
Table 1
Pressure and Gas Force Definitions for Isentropic Compression Cycle
Region Description π(π‘) πΉπ(π‘)
[1] β [2]
Compression Stage - Suction valve closes at BDC [1]
and gas is compressed as piston moves back towards the
head. Chamber pressure increases from suction to
discharge pressure in an isentropic process.
ππ (ππ΅π·πΆπ₯(π‘)
)π
π΄π (ππ (ππ΅π·πΆπ₯(π‘)
)π
β ππ )
[2] β [3] Discharge Stage - Discharge valve opens when
chamber pressure reaches discharge pressure at [2].
Chamber pressure remains constant while valve is open.
ππ π΄π(ππ β ππ )
[3] β [4] Decompression Stage - Discharge valve closes at TDC
[3] and gas is decompressed as piston moves away from
the head. Chamber pressure decreases from discharge to
suction pressure in an isentropic process.
ππ (πππ·πΆπ₯(π‘)
)π
π΄π (ππ (πππ·πΆπ₯(π‘)
)π
β ππ )
[4] β [1] Suction Stage - Suction valve opens when chamber
pressure reaches suction pressure at [4]. Chamber
pressure remains constant while valve is open.
ππ π΄π(ππ β ππ ) = 0
2.2 MECHANICAL SYSTEM MODEL
The mechanical dynamics of this system can be expressed mathematically as follows
πΉπ + πΉπ = ποΏ½ΜοΏ½ + πΆοΏ½ΜοΏ½ + πΎ(π₯ β πΏ0) (2)
πΉπ = πΌπΌ (3)
where πΉπ(πΌ) β β is the motor force, π β β is the mass of the piston, πΆ β β is the coefficient of
friction, πΎ β β is the combined stiffness of the springs connecting the piston to the chassis, π₯(π‘) β β is the
position of the piston with respect to the cylinder head, οΏ½ΜοΏ½(π‘), οΏ½ΜοΏ½(π‘) β β are respectively the first and second
8
derivatives of π₯(π‘), i.e. velocity and acceleration, and πΏ0 β β is the equilibrium position of the piston with
respect to the cylinder head. In (3) πΌ β β is the motor constant and πΌ(π‘) β β is the motor current.
A set of terms are used within this work to refer to per cycle metrics which describe the stroke of the
piston under steady-state conditions. The point during a cycle at which the piston is closest to the cylinder
head is referred to as top dead center and is here denoted as π₯ππ·πΆ β β. The point at which the piston is
farthest from the cylinder head during a cycle is referred to as bottom dead center and is denoted as π₯π΅π·πΆ β
β. The distance traveled between these two points is referred to as the stroke length and is denoted as
βπ₯ππΏ β β which can be defined as
βπ₯ππΏ β π₯π΅π·πΆ β π₯ππ·πΆ . (4)
The point halfway between top and bottom dead center is to as the midstroke and is denoted as π₯πππ β
β. Based on the dynamics of (2) and the fact that πΉπ(π‘) β₯ 0 we can see that increasing gas force has the
effect of increasing the midstroke of the piston away from π₯0.
This system has a frequency at which the mechanical efficiency reaches its maximum, referred to as
the resonant frequency. Since the mechanical output power of the linear motor is a product of the motor
force and the piston velocity, we can see that this power and therefore the system efficiency is maximized
when the force πΉπ(π‘) is in phase with the piston velocity οΏ½ΜοΏ½(π‘). From the dynamics in (2) and (3) we can
see that this phase difference is a function of the system frequency excited by the current πΌ(π‘). For a
standard mass-spring-damper system the resonant frequency is fixed, however, the addition of the nonlinear
gas compression force causes this frequency to change with the thermodynamic load.
2.3 ELECTRICAL SYSTEM MODEL
An equivalent electrical circuit model for the linear motor which drives the compressor piston is
shown in Fig. 3. From this circuit model an electrical dynamic equation can be written as
π£π = πΏππΌΜ + π ππΌ + πΌοΏ½ΜοΏ½ (5)
where π£π(π‘) β β is the voltage applied to the motor, πΌ(Μπ‘) is the derivative of the motor current, πΏπ β
β is the inductance of the motor windings, and π π β β is the resistance of the motor windings. This model
is a simplification of the machine dynamics in that it assumes that all machine parameters are constant. In
9
reality the motor constant πΌ and the inductance πΏπ vary with position π₯(π‘) due to the finite length of the
stator [16].
Ri Li
)(tv a )(tx
)(tI
Figure 3. Equivalent electrical circuit model of a linear motor.
2.4 POWER ELECTRONIC SYSTEM MODEL
For this application a voltage source inverter is used to generate the voltage applied to the motor
π£π(π‘). Figure 4 below shows the h-bridge topology used for this system. This generalized topology shows
a bulk capacitance supporting a DC bus voltage πππ. The source of this DC bus may be a battery, the
output of another converter, or in most cases a rectifier. In this model we assume that the ripple on this DC
bus is negligible and thus that πππ can be treated as a constant.
Figure 4. H-bridge inverter topology.
10
As shown this topology consists of two pairs of series-connected semiconductor switching devices,
each comprising what is referred to as a switching leg. The switches in each leg are driven with
complementary gate signals with appropriate dead-time to prevent shoot-through events, i.e. a case when
both switches are βonβ at the same time, which would result in the DC source being short-circuited.
Through the management of these gate signals each switching leg is capable of outputting a voltage of
Β±πππ at its midpoint. By considering the differential voltage between the two switching leg outputs we can
see that the output voltage π£π(π‘) has three potential voltage states: βπππ , 0, πππ. A pulse width modulation
scheme can then be used to design a duty cycle for each switching leg which achieves a desired AC
reference voltage. A unipolar PWM switching scheme is selected for this application for its superior total
harmonic distortion (THD) over other schemes [22].
Provided that the switching frequency ππ π€ of the PWM is sufficiently higher than the frequency of the
reference AC voltage which is being generated we find that the frequency spectrum of the inverter output
voltage π£π(π‘) is identical to that of the reference AC voltage with additional sidebands starting at 2ππ π€ [22].
Assuming that the inductance of the motor πΏπ is sufficiently high, we can assume that these high frequency
sidebands are effectively filtered and have negligible effect on the system. Thus in our later analysis of the
system electrical dynamics we can ignore the high frequency switching effects and consider that the
inverter output is equal to the reference voltage, i.e. that the inverter is an ideal AC source. Note that this
assumption is valid only if the frequency of π£π(π‘) is sufficiently smaller than ππ π€.
11
3 SIMULATION PLATFORM
The Matlab/Simulink software package was used for simulation of the system model detailed in
Section II to be used for validation of the subsequently developed algorithms. The parameters used for this
model are given in Table 2 below. These parameters reflect the nominal values of the real compressor
utilized in the experimental platform described in the subsequent section. This model includes a C-coded
S-function block which is used to model the highly nonlinear piecewise compression cycle
thermodynamics. The PLECS Blockset was also used to model the h-bridge inverter detailed in Section
2.4. This model uses a DC bus of πππ = 350 [π] and a switching frequency of ππ π€ = 2 [ππ»π§]. This
configuration mirrors the actual inverter utilized in the experimental setup and detailed in the following
section. A variable step ode45 solver was utilized for this simulation model.
Table 2
Linear Vapor Compressor Simulation Model Parameters
Parameter Name Value Units
π π Stator Winding Resistance 6.3 [πΊ]
πΏπ Stator Winding Inductance 0.346 [π»]
πΌ Motor Constant 75.7 [π π΄β ]
π Piston Mass 0.65 [ππ]
πΆ Piston Friction Coefficient 4.5 [π β π πβ ]
πΎ Spring Constant 66,700 [π πβ ]
ππ Discharge Pressure 120 [ππ π]
ππ Suction Pressure 16.6 [ππ π]
π΄π Piston Area 5.3Γ10β4 [π2]
12
π Specific Heat Ratio of Refrigerant 1.07
The majority of algorithms developed and presented in this work have been tested both in simulation
and experimentally. For the sake of brevity, simulation results are only provided in cases where successful
experimental results are unavailable.
13
4 EXPERIMENTAL PLATFORM
4.1 HARDWARE DESIGN
An experimental setup was constructed for real-time implementation of the developed algorithms. A
block diagram of this experimental setup is shown in Figure 5. The compressor was setup to compress air
with the suction valve open to room air. The discharge valve was connected to a nozzle, to adjust the outlet
flow and allow for the buildup of pressure within the system. A pressure gauge was placed in series with
the nozzle to measure of the discharge pressure of the compressor. An H-bridge inverter with a 350 [V] DC
bus provided by an AC/DC converter and a 2 [kHz] unipolar switching scheme [22] with the same topology
shown in Figure 4 was used to generate the motor voltage π£π(π‘) applied to the linear compressor. A LEM
LA25-NP Hall-effect current sensor was used to obtain the current measurement πΌ(π‘). A U.S. Digital
EM1-0-250-I optical encoder with a corresponding LIN-250-1-1 transmissive linear strip was used to
measure the displacement π₯(π‘). This measurement was used as a reference for testing of sensorless control
schemes and also utilized in a number of offline algorithms.
H-BridgeInverter EM1
encoder
MLA25-NP current sensor
cRIO-9022 /cRIO-9113
v a
xIPWM
VdcAC
AC/DC Converter
Figure 5. Block diagram of experimental hardware setup.
A National Instruments CompactRIO device consisting of a cRIO-9022 controller and a cRIO-9113
chassis with onboard Virtex-5 LX50 FPGA was selected for execution of the developed algorithms. NI
14
modules were selected to meet the I/O requirements of the project. An NI-9401 digital output module was
used for generation of the PWM signals which control the inverter switching as well as other digital
communication. An NI-9411 digital input module was used for reading the digital pulse trains generated by
the encoder. Finally, an NI 9215 analog input module was used for measurement of the current sensor
voltage output. A numerical derivative was used to obtain a velocity signal οΏ½ΜοΏ½(π‘) from the measured
displacement π₯(π‘). A 4th order Butterworth filter with cutoff frequency of 200 [Hz] was used to filter the
measured current and velocity signals.
4.2 SOFTWARE DESIGN
In order to execute the variety of control, observation, and estimation schemes developed in this work
a robust and flexible software setup was developed in the LabVIEW development suite. This setup consists
of a hierarchy of three main programs, or virtual instruments (VIs) as they are called in LabVIEW, which
each run on a different hardware target. As mentioned in the previous section the CompactRIO consists of
two parts, a chassis with onboard FPGA and a controller with a real-time processor. The CompactRIO was
setup to run in FPGA mode for maximum performance and functionality, meaning that all I/O are accessed
at the FPGA level. For this reason all data acquisition and algorithms are executed on the FPGA in a
program which we will refer to as the FPGA VI. This FPGA VI is typically run at a sample and loop rate
of 50 [kHz] although faster sample rates were achievable.
Real-time waveform data to be used for visualization is transferred losslessly to the Real-Time VI via
DMA (direct memory access) FIFO (first in, first out) data structures. This Real-Time VI runs on the real-
time processor of the CompactRIO and its primary function is to control data transfer between the FPGA
VI and the Host VI which runs on the host PC. This data transfer includes the waveform data previously
mentioned as well as commands from the Host VI to the FPGA VI and indicator data from the FPGA VI to
the Host VI. Communication between the Real-Time VI and the Host VI is achieved over a hardwired
Ethernet connection.
The primary function of the Host VI is to serve as a user interface (UI) allowing for visualization and
recording of the desired waveform data as well as control over all algorithm parameters. The Real-Time VI
and Host VI were written in such a way that they are easily interfaced with a variety of versions of the
15
FPGA VI. In turn all versions of the FPGA VI, even those implementing vastly different algorithms,
incorporated the same data transfer code to allow them to be used interchangeably. A representation of the
software architecture described is shown in Figure 6 below.
Figure 6. Diagram of software architecture and hardware integration for NI CompactRIO platform.
16
5 PARAMETER ESTIMATION
5.1 ELECTRICAL PARAMETERS
5.1.1 ESTIMATOR DESIGN
An adaptive least-squares estimator is to be developed for estimation of the electrical parameters of
the linear motor model presented in Section 2.3 based on the dynamics in (5). For the proceeding
mathematical analysis the following assumptions are made.
Assumption 1: The parameters π π, πΌ and πΏπ are unknown with π π being a constant and πΌ and πΏπ being
slowly time-varying such that οΏ½ΜοΏ½(π‘) β 0 and οΏ½ΜοΏ½π(π‘) β 0.
Assumption 2: The variables π£π(π‘), πΌ(π‘), πΌ(Μπ‘), π₯(π‘), οΏ½ΜοΏ½(π‘) are all piecewise continuous and bounded
and, with the exception of πΌ(Μπ‘), all are measurable.
Assumption 3: The initial conditions of the variables π£π(π‘0), πΌ(π‘0), πΌ(Μπ‘0), οΏ½ΜοΏ½(π‘0) are all equal to zero;
i.e. estimation begins when the machine is at rest and unexcited.
An adaptive least-squares estimator for the unknown electrical parameters, πΌ, π π , and πΏπ is to be
designed utilizing the electrical dynamic equation (5). To facilitate the estimator development this equation
is rewritten as a linear combination of constants and variables as follows:
π· β πππ (6)
where π·(π‘) β β, π(π‘) β β1Γ3 are defined as follows
π· β [πΌ]Μ (7)
π β [π£π βπΌ βοΏ½ΜοΏ½] (8)
and ππ β β3 represents the unknown parameters to be estimated
ππ β [1
πΏπ π ππΏπ πΌ
πΏπ]π
. (9)
17
To implement the estimator design it is required that all signals in (6) be measureable, but this is not
the case for π·(π‘) due to its dependence on πΌ(Μπ‘). A filtering technique is utilized to overcome this
dependence. The plant of estimation to be used is a linearly stable, strictly proper low-pass filter π»π(π ) β β
defined in the Laplace domain as
π»π(π ) β
πππ + ππ
(10)
where ππ β β+ is a constant gain and π β β is the Laplace variable. This value determines the cutoff
frequency of the filter and should be selected in consideration of noise cancellation versus signal
attenuation. Convolving this filter with the signal π·(π‘) a filtered plant output π·π(π‘) β β can be defined as
follows
π·π(π‘) β βπ(π‘) β π·(π‘) (11)
where βπ(π‘) β β is the inverse Laplace transform of π»π(π ). Substituting (6) into (11) a form of the
filtered signal π·π(π‘) can be written as follows
π·π(π‘) β ππ(π‘)ππ (12)
where ππ(π‘) β β1Γ3 is defined as
ππ(π‘) β βπ(π‘) β π(π‘). (13)
The dependence of π·π(π‘) on πΌ(Μπ‘) can then be eliminated by utilizing the differentiation property of
Laplace transforms as shown below
πΌΜ(π ) = π πΌ(π ) β π(π‘0) (14)
where πΌ(π ) β β is the Laplace transform of πΌ(π‘). Given from Assumption 3 that π(π‘0) = 0, this term can be
eliminated from (14). From this a modified filter π»πβ²(π ) β β can be defined as
π»πβ² (π ) β
πππ
π + ππ
(15)
Utilizing this modified filter, π·π(π‘) can be rewritten in terms of the measurable signal πΌ(π‘) as follows
π·π(π‘) = βπβ² (π‘) β πΌ(π‘) (16)
where ββ²π(π‘) β β is the inverse Laplace transform of π»β²π(π ). The object of the estimator is to update
values of the estimate vector οΏ½ΜοΏ½π according to the error of the estimates. Estimate error is defined as follows
οΏ½ΜοΏ½π β ππ β οΏ½ΜοΏ½π . (17)
18
Due to the fact that ππ is unknown, the error signal οΏ½ΜοΏ½π cannot be directly calculated. However, an
alternate error signal, οΏ½ΜοΏ½π(π‘), can be generated by first substituting the estimate vector οΏ½ΜοΏ½π into (12) to obtain
an estimate of the plant input οΏ½ΜοΏ½π(π‘) as shown
οΏ½ΜοΏ½π β πποΏ½ΜοΏ½π (18)
οΏ½ΜοΏ½π β π·π β οΏ½ΜοΏ½π . (19)
By substituting (12) and (18) into (19) and utilizing (17) it can be seen that this error signal is related
to the actual estimate error as follows
οΏ½ΜοΏ½π = πποΏ½ΜοΏ½π . (20)
From the error signal οΏ½ΜοΏ½π(π‘) an adaptive update rule using the least-squares estimation method can be
developed as follows
οΏ½ΜΜοΏ½π β βππππππ
ποΏ½ΜοΏ½π
1 + πΎππππππππ (21)
where ππ , πΎπ β β+ are constant gains and ππ(π‘) β β
3Γ3 is the covariance matrix defined by the update
equation
οΏ½ΜοΏ½π β βππππππ
πππππ
1 + πΎππππππππ (22)
where ππ(π‘0) = πππΌ3, ππ β β+ is a constant gain, and πΌ3 β β
3Γ3 is the standard identity matrix. These gain
values ππ , πΎπ and ππ directly affect the convergence rate of the system as well as its sensitivity to noise.
The value of οΏ½ΜοΏ½π resulting from the update law in (21) can be used to infer estimates of the unknown
parameters πΌ, π π, and πΏπ by means of the following set of equations
οΏ½ΜοΏ½ =
οΏ½ΜοΏ½π3οΏ½ΜοΏ½π1
, οΏ½ΜοΏ½π =οΏ½ΜοΏ½π2οΏ½ΜοΏ½π1
, οΏ½ΜοΏ½π =1
οΏ½ΜοΏ½π1 (23)
Remark 5.1.1: From (23) it is clear that special care needs to be taken to avoid οΏ½ΜοΏ½π1(π‘) = 0. To
achieve this condition, the projection algorithm described in Section 2.3.1 of [18] must be utilized. This
algorithm takes into account οΏ½ΜοΏ½π1(π‘) and οΏ½ΜΜοΏ½π1(π‘) to keep οΏ½ΜοΏ½π1(π‘) > 0, while maintaining stability and
convergence of the least-squares estimation strategy.
Theorem 5.1.1: The least-squares algorithm described by (21) and (22) ensures that οΏ½ΜοΏ½π(π‘) β 0 as π‘ β
β provided the following sufficient conditions are met: (i) the plant of estimation is strictly proper, (ii) the
19
input is piecewise continuous and bounded, (iii) the output of the plant of estimation is bounded, and (iv)
the following persistence of excitation condition holds
π½1πΌ3 β€ β« ππ
π(π)ππ(π)πππ‘+πΏ
π‘0
β€ π½2πΌ3 (24)
where π½1, π½2, πΏ β β+ are constants and πΌ3 β β
3Γ3 is the standard identity matrix.
Proof: To prove that οΏ½ΜοΏ½π(π‘) β 0 as π‘ β β, Theorem 2.5.3 from [18] is followed directly. Condition (i)
is proved to be valid as the plant of estimation described in (10) can be seen to be strictly proper.
Condition (ii) is met by recognizing that the reference input to the plant π·(π‘) is solely dependent upon the
signal πΌ(Μπ‘) which as stated in Assumption 2 is bounded and piecewise continuous. To prove condition (iii)
is met: by utilizing standard linear analysis tools it can be shown that since π·(π‘) is bounded π·π(π‘), οΏ½ΜοΏ½π(π‘) β
ββ. Condition (iv) is dependent on the way in which the system is excited. As such, it will be shown in
the implementation of this algorithm in Section 5.1.2 that the operating conditions used satisfy this
condition.
5.1.2 EXPERIMENTAL RESULTS
The parameter estimator described in (21) and (22) was implemented on the experimental platform
under the conditions stated. The resulting gas force is highly nonlinear and imparts significant harmonic
content into the mechanical dynamics. This harmonic content is propagated into the electrical dynamic
equation via the velocity in the back EMF term (5), thus fulfilling the persistence of excitation mentioned
in Theorem 5.1.1. Gain values of ππ = 100, ππ = 40, and ππ = 0.2 were selected via trial and error for
optimal convergence and noise sensitivity.
20
Figure 7. Estimation results for the electrical parameters πΌ, π π , πΏπ.
Two different trials were performed using different sets of initial conditions. The first trial used
values of οΏ½ΜοΏ½(π‘0) = 70 [π π΄β ], οΏ½ΜοΏ½π(π‘0) = 5 [πΊ] and οΏ½ΜοΏ½π(π‘0) = 0.4 [π»], while the second used values of
οΏ½ΜοΏ½(π‘0) = 80 [π π΄β ], οΏ½ΜοΏ½π(π‘0) = 8 [πΊ] and οΏ½ΜοΏ½π(π‘0) = 0.2 [π»]. The results of this experiment are shown in
Figure 7. From this we see little to no dependence on initial values for the estimates of πΌ and πΏπ, while a
stronger dependence is seen for π π. Using the average of the convergence values for each trial, the
following parameter estimates were identified for use in the observer algorithms: οΏ½ΜοΏ½ = 75.6 [π π΄β ], οΏ½ΜοΏ½π =
6.47 [πΊ], οΏ½ΜοΏ½π = 0.318 [π»]. A DC resistance test provides further confidence for the resistance estimate with
a result of 6.5 [πΊ].
21
We can make sense of the differences in convergence of the three parameters by considering the
relative magnitudes of the parameter/signal pairs in (6). From (23) we can see that the convergence of οΏ½ΜοΏ½π is
exclusively dependent on οΏ½ΜοΏ½π2 which corresponds to the regressor term π2 from (8). Using the rms values
mentioned previously and the average converged estimator values, an rms of the parameter/signal pair
π2οΏ½ΜοΏ½π2 can be estimated as 10.13 [A/s] while the rms of π1οΏ½ΜοΏ½π1 and π3οΏ½ΜοΏ½π3 are 417.92 [A/s] and 385.13
[A/s], respectively. The difference in the relative magnitude of the term π2οΏ½ΜοΏ½π2 compared to the other terms
in (6) for this excitation explains why weaker convergence is seen for οΏ½ΜοΏ½π2 and therefore οΏ½ΜοΏ½π [18]. Selection
of a more balanced excitation could alleviate this problem, but such an excitation would be of no practical
interest to this application, and since a DC resistance test is easily obtainable there is no motivation to do
so.
5.2 MECHANICAL PARAMETERS
5.2.1 ESTIMATOR DESIGN
The proceeding mathematical analysis requires an additional assumption to be added to those
presented in Section 5.1.1.
Assumption 4: The machine is operating under such conditions that the gas force πΉπ is constant and
equal to zero.
Estimation of the unknown mechanical parameters, π, πΆ, and πΎ utilizing an adaptive least-squares
algorithm is performed utilizing the mechanical dynamic equations (2) and (3). Coupling these two
equations and simplifying them via Assumption 5 the following form of the mechanical dynamics can be
written
πΉ β πππ (25)
where πΉ(π‘) β β, π(π‘) β β1Γ3 are defined as follows
πΉ β [οΏ½ΜοΏ½] (26)
π β [βοΏ½ΜοΏ½ β(π₯ β πΏ0) πΌ] (27)
and ππ β β3 represents the unknown parameters to be estimated
22
ππ β [πΆ
π πΎ
π πΌ
π]π
. (28)
To implement the estimator design it is required that all signals in (25) be measureable, but this is not
the case for πΉ(π‘) due to its dependence on οΏ½ΜοΏ½(π‘). A filtering technique is utilized to overcome this
dependence. The plant of estimation to be used is a linearly stable, strictly proper low-pass filter π»π(π ) β
β defined in the Laplace domain as
π»π(π ) β
πππ + ππ
(29)
where ππ β β+ is a constant gain. This value determines the cutoff frequency of the filter and should be
selected in consideration of noise cancelation versus signal attenuation. Convolving this filter with the
signal πΉ(π‘) a filtered plant output πΉπ(π‘) β β can be defined as follows
πΉπ(π‘) β βπ(π‘) β πΉ(π‘) (30)
where βπ(π‘) β β is the inverse Laplace transform of π»π(π ). Substituting (25) into (30) a form of the
filtered signal πΉπ(π‘) can be written as follows
πΉπ(π‘) β ππ(π‘)ππ (31)
where ππ(π‘) β β1Γ3 is defined as
ππ(π‘) β βπ(π‘) β π(π‘). (32)
The dependence of πΉπ(π‘) on οΏ½ΜοΏ½(π‘) can then be eliminated by utilizing the differentiation property of
Laplace transforms as shown below
οΏ½ΜοΏ½(π ) = π οΏ½ΜοΏ½(π ) β οΏ½ΜοΏ½(π‘0) (33)
where οΏ½ΜοΏ½(π ) β β is the Laplace transform of οΏ½ΜοΏ½(π‘). Given from Assumption 3 that οΏ½ΜοΏ½(π‘0) = 0, this term can
be eliminated from (33). From this a modified filter π»πβ²(π ) β β can be defined as
π»πβ² (π ) β
πππ
π + ππ
(34)
Utilizing this modified filter, πΉπ(π‘) can be rewritten in terms of the measurable signal οΏ½ΜοΏ½(π‘) as follows
πΉπ(π‘) = βπβ² (π‘) β οΏ½ΜοΏ½(π‘) (35)
23
where ββ²π(π‘) β β is the inverse Laplace transform of π»β²π(π ). The object of the estimator is to update
values of the estimate vector οΏ½ΜοΏ½π according to the error of the estimates. Estimate error is defined as
follows
οΏ½ΜοΏ½π β ππ β οΏ½ΜοΏ½π. (36)
Due to the fact that ππ is unknown, the error signal οΏ½ΜοΏ½π cannot be directly calculated. However, an
alternate error signal, οΏ½ΜοΏ½π(π‘), can be generated by first substituting the estimate vector οΏ½ΜοΏ½π into (31) to obtain
an estimate of the plant input οΏ½ΜοΏ½π(π‘) and a corresponding error signal as shown
οΏ½ΜοΏ½π β πποΏ½ΜοΏ½π (37)
οΏ½ΜοΏ½π β πΉπ β οΏ½ΜοΏ½π . (38)
By substituting (31) and (37) into (38) and utilizing (36) it can be seen that this error signal is related to the
actual estimate error as follows
οΏ½ΜοΏ½π = πποΏ½ΜοΏ½π. (39)
From the error signal οΏ½ΜοΏ½π(π‘) an adaptive update rule using the least-squares estimation method can be
developed as follows
οΏ½ΜΜοΏ½π β βππππππ
ποΏ½ΜοΏ½π
1 + πΎππππππππ (40)
where ππ, πΎπ β β+ are constant gains and ππ(π‘) β β3Γ3 is the covariance matrix defined by the update
equation
οΏ½ΜοΏ½π β βππππππ
πππππ
1 + πΎππππππππ (41)
where ππ(π‘0) = πππΌ3, ππ β β+ is a constant gain. These gain values ππ, πΎπ and ππ directly affect the
convergence rate of the system as well as its sensitivity to noise. The value of οΏ½ΜοΏ½π resulting from the update
law in (40) can be used to infer estimates of the unknown parameters π,πΆ, and πΎ by means of the
following set of equations
οΏ½ΜοΏ½ =
1
οΏ½ΜοΏ½π3
πΌ, οΏ½ΜοΏ½π =οΏ½ΜοΏ½π1
οΏ½ΜοΏ½π3
πΌ, οΏ½ΜοΏ½π =οΏ½ΜοΏ½π2
οΏ½ΜοΏ½π3
πΌ (42)
24
Note that the estimates above require knowledge of the motor constant πΌ. This knowledge can be
obtained from the estimate οΏ½ΜοΏ½ in (23), assuming that the electrical parameter estimator has been run
previously.
Remark 5.2.1: From (42) it is clear that special care needs to be taken to avoid οΏ½ΜοΏ½π3(π‘) = 0. To
achieve this condition, the projection algorithm described in Section 2.3.1 of [18] must be utilized. This
algorithm takes into account οΏ½ΜοΏ½π3(π‘) and οΏ½ΜΜοΏ½π3
(π‘) to keep οΏ½ΜοΏ½π3(π‘) > 0, while maintaining stability and
convergence of the least-squares estimation strategy.
Theorem 5.2.1: The least-squares algorithm described by (40) and (41) ensures that οΏ½ΜοΏ½π(π‘) β 0 as π‘ β
β provided the following sufficient conditions are met: (i) the plant of estimation is strictly proper, (ii) the
input is piecewise continuous and bounded, (iii) the output of the plant of estimation is bounded, and (iv)
the following persistence of excitation condition holds
ν1πΌ3 β€ β« ππ
π(π)ππ(π)πππ‘+π
π‘0
β€ ν2πΌ3 (43)
where ν1, ν2, π β β+ are constants and πΌ3 β β
3Γ3 is the standard identity matrix.
Proof: To prove that οΏ½ΜοΏ½π(π‘) β 0 as π‘ β β, Theorem 2.5.3 from [18] is followed directly. Condition
(i) is proved to be valid as the plant of estimation described in (29) can be seen to be strictly proper.
Condition (ii) is met by recognizing that the reference input to the plant πΉ(π‘) is solely dependent upon the
signal οΏ½ΜοΏ½(π‘) which as stated in Assumption 2 is bounded and piecewise continuous. To prove condition (iii)
is met: by utilizing standard linear analysis tools it can be shown that since πΉ(π‘) is bounded πΉπ(π‘), οΏ½ΜοΏ½π(π‘) β
ββ. Condition (iv) is dependent on the way in which the system is excited. As such, it will be shown in
the implementation of this algorithm in Section 5.2.2 that the operating conditions used satisfy this
condition.
5.2.2 SIMULATION RESULTS
The simulation model presented in Section 3 was used to validate the adaptive least-squares algorithm
for the mechanical parameters of the linear compressor. The plant model was modified such that the force
of gas compression was equal to zero per Assumption 4.
25
Because of the lack of gas force in this simulation a different method was required to fulfill the
persistence of excitation mentioned in Theorem 5.2.1. A dual frequency voltage excitation was selected for
this purpose with frequencies of 21 and 49 [Hz] utilized and a peak voltage of 40 [V] for each. Gain values
of ππ = 10, ππ = 50, and ππ = 50 were selected via trial and error for optimal convergence and noise
sensitivity. Initial values of οΏ½ΜοΏ½π(π‘0) = [10 1π5 100]π were used for the estimator. Estimates for the
parameters π, πΆ, and πΎ were calculated according to (42) using the value of πΌ given in Table 2. Fig. 8
shows the results of the simulation and the convergence of all three parameters to the known values used in
the plant model.
Figure 8. Simulation results for adaptive least-squares mechanical parameter estimation scheme.
26
6 REAL-TIME OBSERVERS
6.1 VELOCITY OBSERVER
The goal of this observer is to accurately observe the piston velocity using only the measured motor
current and knowledge of the applied voltage as well as parameter knowledge. From this information a
model-based observer is developed for the piston velocity. Development in the subsequent sections will
show how this observation can be used to obtain valuable operational information such as system
resonance, and ultimately piston position.
6.1.1 OBSERVER DEVELOPMENT
A nonlinear observer is developed for the piston velocity. To facilitate the development of the
velocity observer, the electrical dynamics in (5) can be rewritten as
πΌΜ =1
πΏππ£π β
π ππΏππΌ β ππ (44)
where ππ(π‘) β β is a scaled velocity signal and is defined as follows:
ππ βπΌ
πΏποΏ½ΜοΏ½. (45)
The proceeding mathematical analysis necessitates a new set of assumptions (previous assumptions are
ignored).
Assumption 1: The signals π£π(π‘) and πΌ(π‘) are measurable.
Assumption 2: The variables π£π(π‘), πΌ(π‘), πΌ(Μπ‘), π₯(π‘), οΏ½ΜοΏ½(π‘), ποΏ½ΜοΏ½ (π‘), ποΏ½ΜοΏ½(π‘) are all piecewise continuous and
bounded, hence there exist positive bounding constants ν1π , ν2π β β+ such that |ποΏ½ΜοΏ½(π‘)| < ν1π , |ποΏ½ΜοΏ½(π‘)| < ν2π
Assumption 3: The machine parameters πΏπ , π π , and πΌ are known a priori and are constants with respect
to time.
27
The nonlinear observer to be designed for the signal ππ(π‘) is denoted as ππ(π‘) β β. For a given ππ(π‘),
(44) can be utilized to obtain an observation for πΌ(Μπ‘) defined as
πΌΜΜ β1
πΏππ£π β
π ππΏππΌ β ππ . (46)
Taking the integral of (46) gives the following observation for πΌ(π‘):
πΌ = β« (1
πΏππ£π(π) β
π π
πΏππΌ(π))
π‘
π‘0ππ β β« ππ(π)ππ
π‘
π‘0+ πΌ(π‘0). (47)
Utilizing (46), (47) and the observation ππ(π‘) the following error signals can be defined for the observer:
ππ β ππ β ππ (48)
ππ β πΌ β πΌ (49)
οΏ½ΜοΏ½π = πΌΜΜ β πΌΜ. (50)
Substituting (44) and (46) into (50) and utilizing (48) gives the following relationship between οΏ½ΜοΏ½π(π‘) and
ππ(π‘):
οΏ½ΜοΏ½π = ππ . (51)
A filtered error signal π π(π‘) is defined as
π π β οΏ½ΜοΏ½π + ππ . (52)
From the subsequent stability analysis the following observer is designed for the scaled velocity:
πΜπ β (π1π + 1)π π + π2ππ ππ(ππ) (53)
where π1π , π2π β β+ are constant gains and π ππ(β) is the signum function. The dependence of (53) on
π π(π‘) makes it unrealizable per Assumption 4. A realizable from of οΏ½ΜοΏ½π(π‘) can be obtained by substituting
(52) into (53) and integrating both sides of the resulting equation to obtain the following:
οΏ½ΜοΏ½π= (π1π + 1) [ππ(π‘) β ππ(π‘0) +β« ππ(π)ππ
π‘
π‘0
] + π2πβ« π ππ(ππ(π))ππ
π‘
π‘0
+ οΏ½ΜοΏ½π(π‘0). (54)
Remark 6.1.1: The observer presented in this section may be of particular interest for production-
level applications as it does not require measurement of any of the mechanical states of the compressor.
Remark 6.1.2: The stability analysis presented in the appendix proves that the nonlinear observer
ππ(π‘) defined in (54) converges to the observed signal ππ(π‘). Utilizing (45), a velocity observer οΏ½ΜΜοΏ½π(π‘) can
be written as
28
οΏ½ΜΜοΏ½π βπΏππΌππ . (55)
It is clear from (55) that the accuracy of this observer is directly dependent on the accuracy of the
parameters used.
6.1.2 STABILITY ANALYSIS
Theorem 6.1.1: The scaled velocity observer given in (54) ensures that
ππ(π‘) β ππ(π‘) ππ π‘ β β (56)
provided that the observer gain π2π is selected to meet the following sufficient condition:
π2π > ν1π + ν2π . (57)
Proof: See Appendix.
6.1.3 EXPERIMENTAL RESULTS
The observer described in (54) was implemented on the experimental platform using a current
measurement and knowledge of the applied voltage. Model parameters for the observer were selected based
on the average results of the electrical parameter estimation detailed in Section 5.1.2. A value of π2π =
1.5Γ105 was selected to fulfill the gain condition given in (57) based on the expected range of operating
conditions. A gain of π1π = 4Γ104, which acts similarly to a proportional-integral gain, was selected via
trial and error for optimal convergence and noise sensitivity. Similar performance was seen for values of
π2π > 1Γ105 and π1π > 2Γ104 with upper limits for these gains determined by the variable limits of the
real-time processor to avoid overflow. At π‘ = 0 the observer was initialized.
The results of this experiment are shown in Fig. 9 and 10. Fig. 9 shows the observer error ππ(π‘)
defined in (49) as it converges to approximately zero. Fig. 10 shows the observed velocity οΏ½ΜΜοΏ½π(π‘) derived
via (55) in comparison with the velocity signal οΏ½ΜοΏ½(π‘) obtained from the encoder measurement. Once the
29
observer has converged, the ratio of the rms of the error between the observed and actual velocities, taken
as a percent of the rms of the actual velocity, is calculated as 2.39%. We can see from Fig. 9 and 10 that the
velocity observer is converging much faster than the current observer. From (51) this gives an indication
that οΏ½ΜοΏ½π(π‘) is converging much faster than ππ(π‘).
Figure 9. Convergence of observer error ππ(π‘).
Figure 10. Comparison of velocity, measured οΏ½ΜοΏ½(π‘) and observed οΏ½ΜΜοΏ½π(π‘).
6.2 ACCELERATION OBSERVER
Our objective is to develop an observer for piston acceleration using measurements for piston
velocity, motor current, and parameter knowledge. This observer removes the need for numerical
derivatives which unavoidably amplify the measured noise. Due to its reliance on mechanical
measurements this observer is not suitable for production purposes, however the following subsection will
show how this observer can be modified to remove the need for a piston velocity measurement.
Subsequent sections will also show how the resulting acceleration signal can be used to obtain knowledge
of the absolute piston position.
30
6.2.1 OBSERVER DEVELOPMENT
A nonlinear observer is to be developed to observe piston acceleration while accounting for
uncertainties in the mechanical dynamics. To facilitate the development of the acceleration observer, the
mechanical dynamics described in (2) and (3) can be combined and rewritten as
οΏ½ΜοΏ½ =πΌ
ππΌ + ππ (58)
where ππ(π‘) β β represents the lumped uncertain terms and is defined as follows:
ππ β βπΆ
ποΏ½ΜοΏ½ β
πΎ
ππ₯ +
1
ππΉπ. (59)
The proceeding mathematical analysis necessitates a new set of assumptions (previous assumptions
are ignored).
Assumption 1: The signals πΌ(π‘) and οΏ½ΜοΏ½(π‘) are measurable.
Assumption 2: The variables πΌ(π‘), π₯(π‘), οΏ½ΜοΏ½(π‘), οΏ½ΜοΏ½(π‘), π₯(π‘), ππ(π‘), ποΏ½ΜοΏ½(π‘), ποΏ½ΜοΏ½(π‘) are all piecewise
continuous and bounded, hence there exist positive bounding constants ν1π , ν2π β β+ such that |ποΏ½ΜοΏ½(π‘)| <
ν1π , |ποΏ½ΜοΏ½(π‘)| < ν2π.
Assumption 3: The machine parameters π and πΌ are known a priori and are constants with respect to
time.
The nonlinear observer to be designed for the uncertain terms comprising ππ(π‘) is denoted as ππ(π‘) β
β where the observer is to ensure that ππ(π‘) β ππ(π‘) as π‘ β β. Utilizing ππ(π‘) with (58), an observed
acceleration can be defined as
οΏ½ΜΜοΏ½π βπΌ
ππΌ + ππ. (60)
Taking the integral of (60) gives the following observation for velocity:
οΏ½ΜΜοΏ½π = β«πΌ
ππΌ(π)
π‘
π‘0
ππ + β«ππ(π)ππ
π‘
π‘0
(61)
Utilizing (60), (61) and the observation ππ(π‘) the following error signals can be defined for the
observer:
ππ β ππ β ππ (62)
ππ β οΏ½ΜοΏ½ β οΏ½ΜΜοΏ½π (63)
31
οΏ½ΜοΏ½π = οΏ½ΜοΏ½ β οΏ½ΜΜοΏ½π (64)
Substituting (58) and (60) into (64) and utilizing (62) gives the following relationship between οΏ½ΜοΏ½π(π‘) and
ππ(π‘):
οΏ½ΜοΏ½π = ππ. (65)
A filtered error signal π π(π‘) is defined as
π π β οΏ½ΜοΏ½π + ππ. (66)
From the subsequent stability analysis the following observer is designed for the uncertainty signal:
πΜπ β (π1π + 1)π π + π2ππ ππ(ππ) (67)
where π1π, π2π β π + are constant gains. The dependence of (67) on π π(π‘) makes it unrealizable per
Assumption 7. A realizable from of οΏ½ΜοΏ½π(π‘) can be obtained by substituting (45) into (46) and integrating
both sides of the resulting equation to obtain the following:
ππ = (π1π + 1) [ππ(π‘) β ππ(π‘0) + β« ππ(π)πππ‘
π‘0
] + π2πβ« π ππ(ππ(π))πππ‘
π‘0
+ ππ(π‘0). (68)
Remark 6.2.1: This observer may be of interest for production-level applications as it does not
require measurement of any of the mechanical states of the compressor provided that a velocity observer
such as the one described in Section 4 is used in place of the measured velocity.
Remark 6.2.2: The stability analysis presented in the appendix proves that the nonlinear observer
ππ(π‘) defined in (47) converges to the uncertainty signal ππ(π‘).
6.2.2 STABILITY ANALYSIS
Theorem 6.2.1: The observer defined in (68) ensures that
ππ(π‘) β ππ(π‘) ππ π‘ β β (69)
provided that the observer gain π2π is selected to meet the following sufficient condition:
π2π > ν1π + ν2π . (70)
Proof: See Appendix.
32
6.2.3 EXPERIMENTAL RESULTS
The uncertainty observer described in (68) was implemented on the experimental platform using the
filtered current and velocity measurements described previously. Additionally, a numerical derivative was
used to obtain an acceleration signal from the velocity measurement to be used for comparison. The
nominal mass value listed in Table 2 was utilized for the algorithm along with the motor constant identified
in Section 5.1.2. A value of π2π = 1Γ105 was selected to fulfill the gain condition given in (70) based on
the expected range of operating conditions. A gain of π1π = 2Γ104, which acts similarly to a
proportional-integral gain, was selected via trial and error for optimal convergence and noise sensitivity.
Similar performance was seen for values of π2π > 7.5Γ104 and π1π > 1Γ104 with upper limits for these
gains determined by the variable limits of the real-time processor to avoid overflow.
Figure 11. Comparison of velocity, measured οΏ½ΜοΏ½(π‘) and observed οΏ½ΜΜοΏ½π(π‘).
Figure 12. Comparison acceleration, measured οΏ½ΜοΏ½(π‘) and observed οΏ½ΜΜοΏ½π(π‘).
33
Figure 13. Uncertainty observer ππ(π‘).
Fig. 11-13 show the results of this experiment. Fig. 11 shows that the observed velocity converges to
the actual and that observer error ππ(π‘) is driven to zero. Fig. 12 likewise shows that the observed
acceleration accurately models the measured acceleration with the ratio of the rms error, taken as a
percentage of the rms of the actual acceleration, calculated as 1.21% once the observer has converged. Fig.
13 shows that the uncertainty observer is bounded, validating assertions made in the stability analysis.
6.3 TWO-STAGE OBSERVER
The goal of this observer is to combine the velocity and acceleration observers presented in the
previous two subsections into a single observer which is able to observer acceleration using only a motor
current measurement as well as the parameter knowledge mentioned in the previous two subsections.
6.3.1 OBSERVER DEVELOPMENT
A more practical implementation of the observers in (54) and (68) can be realized by replacing the
velocity measurement required by the uncertainty observer with the output of the velocity observer defined
in (55), running the two observers in parallel. This requires the error signal defined in (63) to be redefined
as ππ β οΏ½ΜΜοΏ½π β οΏ½ΜΜοΏ½π. Unfortunately, it is not possible to prove stability for this implementation.
6.3.2 EXPERIMENTAL RESULTS
This scheme was tested experimentally with the same parameters and gains described previously for
each observer. Once steady-state was achieved both observers were initialized at π‘ = 0. Fig. 14 and 15
34
show the results of this experiment. From these figures we can see that comparable results for boundedness
and convergence are obtained with this scheme as with the previous experiment, validating its use for
practical application. A ratio of the rms error between the measured and observed acceleration, taken as a
percentage of the rms of the actual acceleration, can be calculated as 7.56% once the observers have
converged, showing that there is some loss in accuracy in comparison with the previous implementation
due to the compounding of error between the two observers. There are a number of ways in which this error
could be improved, primarily by improving the accuracy of the velocity observation οΏ½ΜΜοΏ½π. This could be
accomplished by improving the fidelity and resolution of the current and voltage measurements or by
taking into account the time-varying nature of the parameters πΏπ and πΌ.
Figure 14. Velocity observers οΏ½ΜΜοΏ½π(π‘) and οΏ½ΜΜοΏ½π(π‘).
Figure 15. Comparison of acceleration, measured οΏ½ΜοΏ½(π‘) and observed οΏ½ΜΜοΏ½π(π‘).
6.4 GAS FORCE OBSERVER
The goal of this observer is to use measurements for motor current, piston position, and piston
velocity to obtain a real-time observation for the force of gas compression. Due to the multiple mechanical
35
measurements necessary to implement this observer it is not suitable for production purposes. It is
primarily intended as a laboratory tool for model characterization and validation.
6.4.1 OBSERVER DEVELOPMENT
A nonlinear observer is to be developed to observe the force of gas compression. To facilitate the
development of this observer, the mechanical dynamics described in (2) and (3) can be combined and
rewritten as
οΏ½ΜοΏ½ =1
π[πΌπΌ β πΆοΏ½ΜοΏ½ β πΎ(π₯ β πΏ0)] + ππΉ (71)
where ππΉ(π‘) β β represents the scaled gas compression force term and is defined as follows:
ππΉ β1
ππΉπ. (72)
The proceeding mathematical analysis necessitates a new set of assumptions (previous assumptions
are ignored).
Assumption 1: The signals πΌ(π‘), π₯(π‘) and οΏ½ΜοΏ½(π‘) are measurable.
Assumption 2: The variables πΌ(π‘), π₯(π‘), οΏ½ΜοΏ½(π‘), οΏ½ΜοΏ½(π‘), π₯(π‘), ππΉ(π‘), ποΏ½ΜοΏ½(π‘), ποΏ½ΜοΏ½(π‘) are all piecewise
continuous and bounded, hence there exist positive bounding constants ν1πΉ , ν2πΉ β β+ such that |ποΏ½ΜοΏ½(π‘)| <
ν1πΉ , |ποΏ½ΜοΏ½(π‘)| < ν2πΉ.
Assumption 3: The machine parameters π,πΆ, πΎ and πΌ are known a priori and are constants with
respect to time.
The nonlinear observer to be designed for ππΉ(π‘) is denoted as ππΉ(π‘) β β where the observer is to
ensure that ππΉ(π‘) β ππΉ(π‘) as π‘ β β. Utilizing ππΉ(π‘) with (71), an observed acceleration can be defined as
οΏ½ΜΜοΏ½πΉ β1
π[πΌπΌ β πΆοΏ½ΜοΏ½ β πΎ(π₯ β πΏ0)] + ππΉ. (73)
Taking the integral of (73) gives the following observation for velocity:
οΏ½ΜΜοΏ½πΉ =1
πβ«[πΌπΌ β πΆοΏ½ΜοΏ½ β πΎ(π₯ β πΏ0)]
π‘
π‘0
ππ + β«ππΉ(π)ππ
π‘
π‘0
(74)
Utilizing (39), (40) and the observation ππΉ(π‘) the following error signals can be defined for the
observer:
36
ππΉ β ππΉ β ππΉ (75)
ππΉ β οΏ½ΜοΏ½ β οΏ½ΜΜοΏ½πΉ (76)
οΏ½ΜοΏ½πΉ = οΏ½ΜοΏ½ β οΏ½ΜΜοΏ½πΉ (77)
Substituting (71) and (73) into (77) and utilizing (75) gives the following relationship between οΏ½ΜοΏ½πΉ(π‘) and
ππΉ(π‘):
οΏ½ΜοΏ½πΉ = ππΉ. (78)
A filtered error signal π πΉ(π‘) is defined as
π πΉ β οΏ½ΜοΏ½πΉ + ππΉ . (79)
From the subsequent stability analysis the following observer is designed for the uncertainty signal:
πΜπΉ β (π1πΉ + 1)π πΉ + π2πΉπ ππ(ππΉ) (80)
where π1πΉ , π2πΉ β π + are constant gains. The dependence of (80) on π πΉ(π‘) makes it unrealizable per
Assumption 7. A realizable from of ππΉ(π‘) can be obtained by substituting (79) into (80) and integrating
both sides of the resulting equation to obtain the following:
ππΉ = (π1πΉ + 1) [ππΉ(π‘) β ππΉ(π‘0) + β« ππΉ(π)πππ‘
π‘0
] + π2πΉβ« π ππ(ππΉ(π))πππ‘
π‘0
+ ππΉ(π‘0). (81)
Remark 6.4.1: Due to the dependence of this observer on position and velocity measurements, it is
not useful for production-level applications, but may still be of interest for laboratory characterization of
the gas force where full state measurements are available.
Remark 6.4.2: The stability analysis presented in the appendix proves that the nonlinear observer
ππ(π‘) defined in (81) converges to the uncertainty signal ππ(π‘).
6.4.2 STABILITY ANALYSIS
Theorem 6.4.1: The observer defined in (81) ensures that
ππ(π‘) β ππ(π‘) ππ π‘ β β (82)
provided that the observer gain π2π is selected to meet the following sufficient condition:
π2π > ν1π + ν2π . (83)
37
Proof: See Appendix.
6.4.3 EXPERIMENTAL RESULTS
A sinusoidal voltage excitation was applied to the motor with a peak of 194 [V] and a frequency of
62.3 [Hz]. The nozzle was adjusted to bring the discharge pressure of the compressor to 120 [psi]. The
nominal mass, friction coefficient, and spring constant values listed in Table 2 were utilized for the
algorithm along with the motor constant identified in Section 5.1.2. For the observer gain values of π1πΉ =
2Γ104, π2πΉ = 1Γ105 were selected to meet the requirements stated in (83). With the system operating at
steady-state the observer algorithm was turned on at π‘ = 0. Figures 16 and 17 show the results of the
experiment. Fig. 16 shows that the observed velocity converges to the measured and that observer error
ππΉ(π‘) is driven to zero. Fig. 17 shows that the observed gas force is bounded. Without some sort of sensor
to measure gas force we have no reference signal with which to determine accuracy.
Figure 16. Comparison of actual velocity οΏ½ΜοΏ½(π‘) and observed velocity οΏ½ΜΜοΏ½πΉ(π‘).
Figure 17. Observed gas force οΏ½ΜοΏ½π(π‘).
38
6.5 POSITION AND PRESSURE OBSERVER
The goal of this observer is to obtain real-time observations for both the absolute piston position as
well as the suction and discharge pressures. Through this development several other signals are also
incidentally observed such as the gas compression force and the piston acceleration. Whereas the observers
developed for these signals in sections 6.2 and 6.4, respectively, assumed bounded time-varying
uncertainties for the unmodeled terms, this observer development seeks to fully model all terms in the
mechanical dynamics and reduce all uncertainties to parametric uncertainties. With an observer form
which has only parametric uncertainties an adaptive update law can be designed to account for these.
6.5.1 RELATIVE POSITION OBSERVER
Looking at the mechanical dynamic equations (2) and (3) we can see that the acceleration is
dependent upon the absolute piston position π₯(π‘) via the spring force term. This feedback gives us an
opportunity to observe the position via its effects on the mechanical dynamics. Assuming that we have an
accurate velocity observation from Section 6.1, it is natural to approximate the absolute position by
integrating this signal. Unfortunately, this method is not reliable for obtaining an absolute position
observation in practical implementation. This is due to the fact that an artificial DC bias is propagated into
the velocity observation from the input voltage bias of the non-ideal analog to digital converter (ADC)
which is used to obtain the current measurement πΌ(π‘). Looking at (46), we can see that in order to satisfy
πΌΜΜ(π‘) β πΌΜ(π‘) the scaled velocity observer ππ(π‘) must assume a DC offset to cancel out the bias present in the
πΌ(π‘) term of the same equation. This DC offset propagates into the velocity observation per the definition
in (55).
Because of this offset in οΏ½ΜΜοΏ½π(π‘) we can see that the observed position defined in (84) will become
positively or negatively unbounded depending on the sign of the DC offset. In reality, even a measured
velocity will have some finite inaccuracy which over time will cause drift in the integrator. Though this
offset may be measured and compensated for it can never be fully cancelled out without some form of high
39
pass filtering. High pass filtering would have the undesirable effect of removing any real DC component
from the velocity signal which we know occurs during transient periods where the gas compression force,
πΉπ(π‘) is changing the midstroke of the piston.
Thus to make use of this integrated signal we must find a way to artificially bound it. A method is
proposed whereby the integral is reset to a value of zero each time the piston reaches top dead center π₯ππ·πΆ.
As top dead center represents the minimum point in the piston position we can identify this point in the
cycle by looking at when the velocity signal experiences a positive-going zero cross. The time of the most
recent of these occurrences is denoted as οΏ½ΜοΏ½ππ·πΆ. A relative position signal οΏ½Μ οΏ½(π‘) defined using this periodic
resetting can thus be defined as
οΏ½Μ οΏ½(π‘) β β« οΏ½ΜΜοΏ½π(π)ππ‘
οΏ½ΜοΏ½ππ·πΆπ. (84)
We can see that by resetting the integrator to zero at each οΏ½ΜοΏ½ππ·πΆ the resulting signal οΏ½Μ οΏ½(π‘) will have a
minimum value of zero. This methodology assumes that the bias accumulated over each cycle from the
velocity is negligible and does not result in significant discontinuities at the time of resetting. From this
assumption we can treat the signal οΏ½Μ οΏ½(π‘) as a shifted version of the absolute piston position π₯(π‘), where the
value by which the signal has been shifted is equal to π₯ππ·πΆ.
οΏ½Μ οΏ½(π‘) = π₯(π‘) β π₯ππ·πΆ (85)
Assuming that π₯ππ·πΆ is slowly time-varying, which is a valid assumption under steady-state conditions,
gives us the ability to use adaptive methods to estimate this value.
6.5.2 LINEARLY PARAMETERIZED GAS FORCE
In order to fully model the mechanical dynamics we also require a structured form of the unknown
time-varying gas force πΉπ(π‘). Fortunately, we have such a form given in Table 1. Unfortunately, many of
the parameters within this piecewise equation are unknown. However, we can rewrite this equation in a
linearly parameterized form such that the gas force πΉπ(π‘) is equal to the inner product of a vector of known
time-varying regressors π(π‘) β β1Γ2 and a vector of unknown constant parameters π β β2Γ1 as shown:
πΉπ = π΄πππ. (86)
40
From this we can then use adaptive methods to estimate the unknown parameters comprising π. In this
case π consists of the unknown constant suction and discharge pressures, i.e. π β [ππ ππ]π, and the
piecewise regressor terms comprising π(π‘) β [π1 π2] are defined in Table 3 below.
Table 3
Regressor Variable Definitions for Piecewise Gas Force
Stage Piecewise Condition π1 π2
Compression
[1] β [2]
οΏ½ΜοΏ½ < 0,
π(π‘) < ππ
[(π₯π΅π·πΆπ₯(π‘)
)π
β 1] 0
Discharge
[2] β [3]
οΏ½ΜοΏ½ < 0,
π(π‘) β₯ ππ
β1 1
Decompression
[3] β [4]
οΏ½ΜοΏ½ > 0,
π(π‘) > ππ
β1 (π₯ππ·πΆπ₯(π‘)
)π
Suction
[4] β [1]
οΏ½ΜοΏ½ > 0,
π(π‘) β€ ππ
0 0
We can see that by substituting this definition of π(π‘) into (86), we obtain the same definition for gas
force πΉπ(π‘) given in Table 1.
6.5.3 OBSERVER DEVELOPMENT
An observer is to be designed for the mechanical dynamics of the compressor. To facilitate the
development of this observer, the mechanical dynamics described in (2) and (3) can be combined and
substitutions made using (85) and (86) to obtain the following:
οΏ½ΜοΏ½ =1
π[πΌπΌ + π΄πππ β πΆοΏ½ΜοΏ½ β πΎ(οΏ½Μ οΏ½ + π₯ππ·πΆ β πΏ0)] (87)
The subsequent development necessitates the following set of assumptions for this model:
Assumption 1: The signals πΌ(π‘), and οΏ½ΜοΏ½(π‘) are known.
Assumption 2: The variables πΌ(π‘), π₯(π‘), οΏ½ΜοΏ½(π‘), πΉπ(π‘) are all piecewise continuous and bounded.
41
Assumption 3: The machine parameters π,πΆ, πΎ, π΄π, π and πΌ are known a priori and are constants with
respect to time.
Assumption 4: The unknown parameters ππ , ππ , π₯ππ·πΆ are bounded and slowly time-varying such that
οΏ½ΜοΏ½π , οΏ½ΜοΏ½π , οΏ½ΜοΏ½ππ·πΆ β 0.
As mentioned in Assumption 1 a velocity signal οΏ½ΜοΏ½(π‘) is required. It is to be understood that this
signal can be replaced with the observer οΏ½ΜΜοΏ½π(π‘) from Section 6.1 to make the resulting observer sensorless
in a similar way as was done in Section 6.3 with the acceleration observer.
To avoid the need for a position measurement the subsequent development uses the relative position
οΏ½Μ οΏ½(π‘) defined in (84) instead, accounting for the fact that it is shifted from the absolute position. From this
the following error signal can be defined for our observer οΏ½ΜοΏ½(π‘):
οΏ½ΜοΏ½ β οΏ½Μ οΏ½ β οΏ½ΜοΏ½ (88)
Substituting (85) into (88) and taking the time derivative, noting Assumption 4, we obtain the following
velocity error signal for the observer:
οΏ½ΜΜοΏ½ = οΏ½ΜοΏ½ β οΏ½ΜΜοΏ½ (89)
To facilitate the development a filtered error signal π(π‘) β β is defined as
π β οΏ½ΜΜοΏ½ + π1οΏ½ΜοΏ½ (90)
where π1 β β is a positive filter gain. Note that because οΏ½ΜοΏ½(π‘) is known, both οΏ½ΜΜοΏ½(π‘) and π(π‘) are realizable
signals. Taking the time derivative of (90) and substituting in (87) we obtain the following open loop error
dynamics:
οΏ½ΜοΏ½ =1
π[πΌπΌ + π΄πππ β πΆοΏ½ΜοΏ½ β πΎ(οΏ½Μ οΏ½ + π₯ππ·πΆ β πΏ0)] β οΏ½ΜΜοΏ½ + π1οΏ½ΜΜοΏ½. (91)
Motivated by the form of (91) and the subsequent stability analysis the observer οΏ½ΜΜοΏ½(π‘) is designed as
οΏ½ΜΜοΏ½ β1
π[πΌπΌ + π΄πποΏ½ΜοΏ½ β πΆοΏ½ΜοΏ½ β πΎ(οΏ½Μ οΏ½ + οΏ½ΜοΏ½ππ·πΆ β πΏ0)] + π1οΏ½ΜΜοΏ½ + οΏ½ΜοΏ½ + π2π (92)
where π2 β β is a positive control gain, and οΏ½ΜοΏ½(π‘), οΏ½ΜοΏ½ππ·πΆ(π‘) β β are parameter estimates defined by the
following adaptive update laws:
οΏ½ΜΜοΏ½ βπ΄π
ππ€πππ (93)
οΏ½ΜΜοΏ½ππ·πΆ β β
ππ₯πΎ
ππ
(94)
42
where π€ β β2Γ2 is a positive diagonal gain matrix defined as π€ β πππΌ2 (ππ β β being a positive gain) and
ππ₯ β β is a positive estimator gain. Error terms for these estimates are defined as
οΏ½ΜοΏ½ β π β οΏ½ΜοΏ½ (95)
οΏ½ΜοΏ½ππ·πΆ β π₯ππ·πΆ β οΏ½ΜοΏ½ππ·πΆ . (96)
Substituting (92) into (91) we obtain the following closed loop error dynamics for the proposed observer:
οΏ½ΜοΏ½ = βπ2π β οΏ½ΜοΏ½ +1
π[π΄πποΏ½ΜοΏ½ β πΎοΏ½ΜοΏ½ππ·πΆ]. (97)
Remark 6.5.1: The proposed implementation requires knowledge of π(π‘) per the definitions in
(92),(93). However, we can see from Table 3 that π(π‘) depends upon the terms π₯(π‘), π₯ππ·πΆ , π₯π΅π·πΆ , which
are not available in a sensorless implementation. As such, for a sensorless implementation we are forced to
substitute observed values for each of these terms. Motivated by (85) we substitute οΏ½Μ οΏ½(π‘) + οΏ½ΜοΏ½ππ·πΆ for π₯(π‘),
οΏ½ΜοΏ½ππ·πΆ for π₯ππ·πΆ, and οΏ½Μ οΏ½(οΏ½ΜοΏ½π΅π·πΆ) + οΏ½ΜοΏ½ππ·πΆ, where οΏ½ΜοΏ½π΅π·πΆ is the most recent time at which the piston reached bottom
dead center as determined by the negative-going zero-crossing of the velocity signal. Moreover, the
piecewise conditions listed in Table 3 are also dependent upon the unknown pressures. In this case as well
we are forced to use the estimated pressures οΏ½ΜοΏ½π , οΏ½ΜοΏ½π resulting from οΏ½ΜοΏ½(π‘) in place of the actual pressures
ππ , ππ . From the relationship between π(π‘) and πΉπ(π‘) given in (1) and Table 1 we can define the following
observer οΏ½ΜοΏ½(π‘) to be used in the piecewise conditions for π(π‘):
οΏ½ΜοΏ½(π‘) β (π1 + 1)οΏ½ΜοΏ½π +π2οΏ½ΜοΏ½π . (98)
With these substitutions we introduce a recursive feedback into the observer. As this feedback is not
modeled in the development presented in this section the subsequent stability analysis cannot be used to
prove that the closed loop system is stable under these substitutions. Thus it is left to experimental
validation to show whether the proposed sensorless method is stable and convergent.
6.5.4 STABILITY ANALYSIS
Theorem 6.5.1: The closed loop system defined by the observer in (92) and estimators in (93), (94)
together ensure that the observer errors οΏ½ΜοΏ½(π‘), π(π‘) β 0 as π‘ β β.
Proof: A non-negative Lyapunov function π(π‘) is defined for the closed loop system as
43
π β1
2οΏ½ΜοΏ½2 +
1
2π2 +
1
2οΏ½ΜοΏ½ππ€β1οΏ½ΜοΏ½ +
1
2ππ₯οΏ½ΜοΏ½ππ·πΆ2 . (99)
Taking the time derivative of (99) we obtain the following:
οΏ½ΜοΏ½ = οΏ½ΜοΏ½οΏ½ΜΜοΏ½ + ποΏ½ΜοΏ½ + οΏ½ΜοΏ½ππ€β1οΏ½ΜΜοΏ½ +1
ππ₯οΏ½ΜοΏ½ππ·πΆ οΏ½ΜΜοΏ½ππ·πΆ . (100)
Then solving (90) for οΏ½ΜΜοΏ½(π‘) and substituting this and the closed loop error dynamics from (97) into (100),
and also substituting in the time derivatives of (95), (96), taking into account that οΏ½ΜοΏ½π , οΏ½ΜοΏ½π , οΏ½ΜοΏ½ππ·πΆ β 0, we
obtain:
οΏ½ΜοΏ½ = οΏ½ΜοΏ½(π β π1οΏ½ΜοΏ½) + π (βπ2π β οΏ½ΜοΏ½ +1
π[π΄πποΏ½ΜοΏ½ β πΎοΏ½ΜοΏ½ππ·πΆ]) β οΏ½ΜοΏ½
ππ€β1οΏ½ΜΜοΏ½ β
1
ππ₯οΏ½ΜοΏ½ππ·πΆ οΏ½ΜΜοΏ½ππ·πΆ .
(101)
Simplifying (101) and substituting into it (93) and (94) we obtain the following:
οΏ½ΜοΏ½ = βπ1οΏ½ΜοΏ½2 β π2π
2 +1
ππ(π΄πποΏ½ΜοΏ½ β πΎοΏ½ΜοΏ½ππ·πΆ) β
π΄π
πποΏ½ΜοΏ½π β
πΎ
ποΏ½ΜοΏ½ππ·πΆπ. (102)
Simplifying (102) we obtain the final result
οΏ½ΜοΏ½ = βπ1οΏ½ΜοΏ½2 β π2π
2. (103)
From the form of (99) and (105) it is clear that οΏ½ΜοΏ½(π‘), π(π‘), οΏ½ΜοΏ½(π‘), οΏ½ΜοΏ½ππ·πΆ(π‘) β ββ and that οΏ½ΜοΏ½(π‘), π(π‘) β
ββββ2. From (90) and the fact that οΏ½ΜοΏ½(π‘), π(π‘) β ββ we can see that οΏ½ΜΜοΏ½(π‘) β ββ.
To prove boundedness of οΏ½ΜοΏ½(π‘) we must first show that π(π‘) β ββ. This analysis will consider
only cases where π₯(π‘) β₯ 0. Cases where π₯(π‘) < 0 require a modification of the algorithm which is
discussed in Section 6.5.7. Looking at the individual terms π1,π2 we can see that by definition both terms
are constants during all stages of the compression cycle, except for π1 during compression and π2 during
decompression. Looking at π1 during compression we can see that as π₯(π‘) β 0, π1(π‘) β β. However,
from (98) we can see that as π1(π‘) β β, οΏ½ΜοΏ½(π‘) β β as well, meaning that the piecewise condition οΏ½ΜοΏ½(π‘) β₯
οΏ½ΜοΏ½π will always be met before π₯(π‘) = 0 for bounded οΏ½ΜοΏ½π. From Assumption 4 and the fact that οΏ½ΜοΏ½(π‘) β ββ
we can see that οΏ½ΜοΏ½π , οΏ½ΜοΏ½π β ββ. Thus π1(π‘) will always transition into the discharge stage before π₯(π‘) = 0,
giving it an upper bound of οΏ½ΜοΏ½π/οΏ½ΜοΏ½π . For π2 during decompression we again see that the term is problematic
as π₯(π‘) β 0. However if we consider the assumption that π₯(π‘) β₯ 0, then π₯(π‘) can equal zero if and only if
it is at a local minimum, meaning that π₯ππ·πΆ = 0. Substituting this value into the definition of π2 and
taking the limit as π₯(π‘) β 0 we see that π2 = 0. For any non-zero value of π₯(π‘) we can see that π2 is
44
bounded so long as π₯ππ·πΆ is bounded, which was given in Assumption 4. Thus we have shown that
π1,π2 β ββ.
From (97) and the fact that οΏ½ΜοΏ½(π‘), π(π‘), οΏ½ΜοΏ½(π‘), οΏ½ΜοΏ½ππ·πΆ(π‘),π(π‘) β ββ we can see that οΏ½ΜοΏ½(π‘) β ββ.
Since οΏ½ΜοΏ½(π‘), π(π‘) β ββββ2 and οΏ½ΜΜοΏ½(π‘), οΏ½ΜοΏ½(π‘) β ββ Barbalatβs Lemma [23] can be utilized to prove that
οΏ½ΜοΏ½(π‘), π(π‘) β 0 as π‘ β β.
6.5.5 EXPERIMENTAL RESULTS
The model parameters listed in Table 2 were utilized for this experiment along with the following
observer gains and initial values: π1 = 100, π2 = 1000, ππ = 400, ππ₯ = 0.05, οΏ½ΜοΏ½π (π‘0) = 20 [ππ π], οΏ½ΜοΏ½π(π‘0) =
50 [ππ π], οΏ½ΜοΏ½ππ·πΆ(π‘0) = 4 [ππ]. The system was driven under current control with a fixed amplitude of
0.6 [π΄] and a driving frequency of 45.7 [π»π§]. The scheme was implemented in a fully sensorless manner,
meaning that the velocity signal used in (84) and (89) comes from the sensorless velocity observer
developed in Section 6.1.
Figures 18-20 show the transient performance of the observer signals οΏ½ΜοΏ½(π‘), οΏ½ΜΜοΏ½(π‘) during the initial
observer convergence. We can see from Figures 18 and 19 that within two cycles both of these signals
converge to their respective reference signals οΏ½Μ οΏ½(π‘), οΏ½ΜοΏ½(π‘) and thus in Figure 20 that the observer error
signals οΏ½ΜοΏ½(π‘), οΏ½ΜΜοΏ½(π‘), π(π‘) are quickly regulated. We see that there is some residual component of the error
π(π‘) owing to the fact that the parameter estimates οΏ½ΜοΏ½ππ·πΆ , οΏ½ΜοΏ½π , οΏ½ΜοΏ½π have not fully converged within this time
period.
Figures 21 and 22 show the convergence of these parameter updates, which take place over a much
longer time scale. From these figures we can see that not only are these parameters convergent, but that
they display a high level of accuracy when compared to their measured counterparts. In Figure 23-25 we
can see the performance of the observer after the parameters have converged and the observer is operating
at steady-state. Figure 23 shows the total position observer consisting of the relative position observer οΏ½Μ οΏ½(π‘)
and the top dead center estimate οΏ½ΜοΏ½ππ·πΆ. This absolute position observer is shown in comparison with the
actual measured piston position π₯(π‘). From this we can see the high level of accuracy achieved by the
45
adaptive observer. Figures 24 and 25 show the steady-state results of the pressure and gas force estimation.
In Figure 24 we see the instantaneous chamber pressure observer οΏ½ΜοΏ½(π‘) derived from (99) along-side the
pressure estimates οΏ½ΜοΏ½π , οΏ½ΜοΏ½π, showing the movement of the observer through the four compression cycle
stages. Figure 25 shows the resulting gas force observer οΏ½ΜοΏ½π(π‘). Unfortunately, since measurements of
internal pressures and forces are difficult to obtain we have no means of judging the accuracy of the two
observer signals οΏ½ΜοΏ½(π‘) and οΏ½ΜοΏ½π(π‘).
Figure 18. Comparison of relative position signal οΏ½Μ οΏ½(π‘) and observer position signal οΏ½ΜοΏ½(π‘) during initial
transient.
Figure 19. Comparison of velocity signal οΏ½ΜοΏ½(π‘) and observer velocity οΏ½ΜΜοΏ½(π‘) during initial transient.
46
Figure 20. Convergence of observer error signals οΏ½ΜοΏ½(π‘), οΏ½ΜΜοΏ½(π‘), π(π‘) during initial transient.
Figure 21. Convergence of parameter estimate οΏ½ΜοΏ½ππ·πΆ to actual top dead center value π₯ππ·πΆ.
Figure 22. Convergence of parameter estimates οΏ½ΜοΏ½π , οΏ½ΜοΏ½π to actual pressures ππ , ππ .
47
Figure 23. Comparison of position observer including top dead center estimate (οΏ½Μ οΏ½(π‘) + οΏ½ΜοΏ½ππ·πΆ) with actual
absolute position π₯(π‘) under observer steady-state.
Figure 24. Instantaneous chamber pressure observer οΏ½ΜοΏ½(π‘) during observer steady-state, shown with
estimated suction and discharge pressures οΏ½ΜοΏ½π , οΏ½ΜοΏ½π.
Figure 25. Instantaneous gas force observer οΏ½ΜοΏ½π(π‘) during observer steady-state as calculated from
estimated pressures οΏ½ΜοΏ½π , οΏ½ΜοΏ½π.
48
6.5.6 INCOMPLETE CYCLE MODIFICATION
The gas force model presented in Section 2.1 and utilized in Section 6.5.2 is based upon the
assumption that the compression cycle passes through all four stages in order: compression, discharge,
decompression, suction. It is this assumption that allows us to assume that the chamber pressure π(π‘) at
the beginning of the compression stage is equal to ππ and thus relate the gas force to ππ via the π1 term for
that stage. Similarly, we assume that at the beginning of the decompression stage that π(π‘) is equal to ππ,
which allows us to relate the gas force to ππ via the π2 term for that stage.
However, during startup and shutdown conditions it is likely that the piston stroke is not large enough
to reach the piecewise pressure conditions necessary to enter the discharge and suction stages of the
compression cycle resulting in an incomplete cycle. For instance, if π(π‘) does not exceed ππ during the
compression stage then the system will skip the discharge stage and move directly into the decompression
stage, but instead of beginning that stage at ππ, it will begin it at whatever pressure the compression stage
ended. Similarly, if π(π‘) does not decrease to ππ during the decompression stage then the system will skip
the suction stage and move directly into the compression stage starting at whatever pressure the
decompression stage ended instead of ππ .
For both of these cases the algorithm must be modified to reflect this reality, otherwise the resulting
π(π‘) and πΉπ(π‘) will be discontinuous at these stage transitions causing instability in the observer. To
remedy this, whenever the observer enters either the compression or decompression stage it checks to see
which stage it just left. If this reveals that a discharge or suction stage was skipped then the algorithm uses
the last value of π(π‘) in place of ππ or ππ (for compression and decompression stages, respectively) in the π
vector. Also, since π(π‘) is no longer dependent on ππ or ππ during this stage, the associated update οΏ½ΜΜοΏ½ is
forced to be zero for the duration of the stage. This is meant to prevent runaway conditions where the
estimator would attempt to reduce error π(π‘) by updating οΏ½ΜοΏ½π or οΏ½ΜοΏ½π via (93), (94), but since the respective
term is no longer affecting the estimated gas force, there is no feedback whereby it can reduce the error,
causing it to runaway during that time.
These modifications are relatively easy to implement, but are difficult to express in a closed form, and
thus cannot be proven to maintain stability via the analysis given in Section 6.5.4.
49
6.5.7 SOFT CRASH MODIFICATION
Another instance in which the gas force model presented in Section 2.1 is violated is under the soft
crash condition. Soft crashing is the result of the compressor piston being overdriven, causing it to come
into physical contact with the discharge valve as shown in Figure 26. Here, the piston shown in red,
extends past the end of the compression cylinder, shown in green, causing it to come into contact with the
discharge valve shown in grey.
Figure 26. Example of soft crash condition for linear compressor.
The cylinder head, which corresponds to the far left extremity of the compression cylinder shown in
Figure 26, is defined as the reference position (π₯ = 0) of the piston because at this point the compression
chamber volume is theoretically equal to zero. Thus when the piston extends past this point the piston
position becomes negative. Given the model from Section 2.1 this would imply a negative volume, which
is of course impossible, we are forced to modify the model and the subsequent observer design to ensure
accurate modeling of soft crash conditions.
From Figure 26 we can see that the presence of the discharge valve causes the volume of the
compression chamber to remain equal to zero until the piston has withdrawn past the head and come out of
contact with the discharge valve. We can also see that because the pressure on the back-side of the
discharge valve is by definition ππ, that during this period the effective chamber pressure π(π‘) (as it effects
the gas force) will still be equal to ππ, even though technically there is no gas in the compression chamber.
Another important fact to note is that in order for the compression volume to reach zero, as it must before
50
soft crashing, the chamber pressure will by necessity exceed ππ. We can infer this from the form of
chamber pressure expression given for the compression stage in Table 1. Since π₯(π‘) is in the denominator
of this expression, we can see that as π₯(π‘) β 0, π(π‘) β β. Thus any finite discharge pressure ππ will be
exceeded before π₯(π‘) = 0, resulting in transition to the discharge stage of the cycle. This means that the
stage of the cycle at the time of soft crashing will always be the discharge stage.
Therefore, since the compressor is in discharge stage before soft crashing, and since the effective
chamber pressure during soft crashing is ππ, which is the same as in the discharge stage, we can accurately
model the effect of soft crashing on gas force by keeping the compressor in the discharge stage during this
condition. This can be easily done by redefining the piecewise condition which determines transition from
the discharge to the decompression stage. By forcing the model to wait until π₯ > 0 before beginning
decompression we are able to obtain the desired effect. One point that remains to be defined is the value of
π₯ππ·πΆ to be used in the π2 expression in Table 3 during the decompression stage. As previously defined,
π₯ππ·πΆ corresponds to the minimum position of the piston. However, since during soft crashing this is a
negative value and would thus imply negative volume, we need to make some modification. Forcing π₯ππ·πΆ
to be equal to zero in this case makes sense as this would reflect the reality that the volume is beginning the
decompression stage at zero, however from the decompression expression for π2 we see that this would
cause π2 to be zero, meaning that π(π‘) would immediately transition from ππ to zero, causing transition
immediately to the suction stage. While this is not necessarily incorrect for a theoretical model, to give the
observer continuity a small positive constant ν β β is utilized instead, resulting in a very short but still
continuous decompression stage. In order to prevent conditions where π2 > 1 during decompression, we
also use ν as the boundary condition (instead of π₯ = 0) for entering the decompression stage. A revised
version of the regressor definitions given in Table 3 with piecewise conditions which have been modified
to account for soft crash conditions is shown in Table 4 below. We can see that these modified definitions
of π1,π2 maintain the boundedness required by the stability analysis.
Table 4
Revised Regressor Variable Definitions for Piecewise Gas Force
51
Stage Piecewise Condition π1 π2
Compression
[1] β [2]
οΏ½ΜοΏ½ < 0 | π₯ β€ ν,
π(π‘) < ππ
[(π₯π΅π·πΆπ₯(π‘)
)π
β 1] 0
Discharge
[2] β [3]
οΏ½ΜοΏ½ < 0 | π₯ β€ ν,
π(π‘) β₯ ππ
β1 1
Decompression
[3] β [4]
οΏ½ΜοΏ½ > 0 & π₯ > ν
π(π‘) > ππ
β1 (π₯ππ·πΆπ₯(π‘)
)π
Suction
[4] β [1]
οΏ½ΜοΏ½ > 0 & π₯ > ν,
π(π‘) β€ ππ
0 0
52
7 PER-CYCLE OBSERVATIONS
To achieve the system level control objectives of resonance-tracking and stroke control it is necessary
to derive auxiliary observations of system-level states from the instantaneous observer signals obtained in
Sections 6. Specifically, two per-cycle signals are needed for the desired control, the relative phase
between velocity and current, and the piston top dead center value π₯ππ·πΆ. A method for estimating π₯ππ·πΆ has
already been provided in Section 6.5, however for thoroughness an alternate derivation based on earlier
methodologies is presented in Section 7.4. This method requires observations for the piston stroke length
βπ₯ππΏ and bottom dead center π₯π΅π·πΆ which are derived in the preceding sections. The accuracies of these
observations is directly related to the accuracy of the two-stage observer presented in Section 6.4 which has
already been demonstrated in Section 6.4.3. Similarly, the phase observation derived in Section 7.1 is
directly related to the velocity observer developed in Section 6.1. As such independent experimental
validations are not provided for these per-cycle observations, however, the results in Section 8.5, which
utilize these observations can be referenced for this purpose.
7.1 PHASE OBSERVATION
Per the definition of resonance given in Section II the relative phase difference between the piston
velocity and the motor force, which per (3) has the same phase as the motor current, can be used as an
indicator of whether the system is operating at its resonance frequency [14], [15]. Utilizing the observed
velocity from (55), an observed phase can be obtained by comparing the zero-crossing of this signal with
that of the measured current. At the time of a positive zero-crossing in οΏ½ΜΜοΏ½π(π‘), the elapsed time since the
most recent positive zero-crossing in πΌ(π‘) is calculated and denoted as βπ‘. From βπ‘ a phase observation οΏ½ΜοΏ½ππΌ
in degrees can be defined as
οΏ½ΜοΏ½ππΌ β 360 β βπ‘ β π (104)
53
where π β β is the fundamental frequency of the system. Standard phase wrapping can be used to shift the
domain of οΏ½ΜοΏ½ππΌ from [0Β°, 360Β°) to (β180Β°, 180Β°], which is more desirable for control purposes. Per this
measurement methodology a new phase observation is obtained once per cycle. It should be noted that this
calculation of relative phase is used for ease of implementation as zero-crossings are easily detectable. In
reality, because of the nonlinear gas force, the velocity and current signals will contain multiple harmonics
each with their own relative phase.
7.2 STROKE LENGTH OBSERVATION
As defined in (4), the stroke length is equal to the distance traveled between top dead center (the
minimum position of the piston stroke) and bottom dead center (the maximum position of the piston
stroke). Given that the relative position signal οΏ½Μ οΏ½(π‘) defined in (85) is equal to zero at top dead center, we
can see that at bottom dead center it will be equal to the stroke length of the observed position. Bottom
dead center time can be identified by the negative-going zero-crossing of the observed velocity οΏ½ΜΜοΏ½π(π‘),
similar to how the timing of top dead center was identified in Section 6.5.1. The time of the most recent of
these occurrences is denoted as οΏ½ΜοΏ½π΅π·πΆ. Thus observed stroke length can be defined as
βοΏ½ΜοΏ½ππΏ β οΏ½Μ οΏ½(οΏ½ΜοΏ½π΅π·πΆ). (105)
From this we can see that a new value for βοΏ½ΜοΏ½ππΏ is obtained only once per cycle.
7.3 BOTTOM DEAD CENTER OBSERVATION
A scheme similar to the one proposed in [6] is used for estimation of the bottom dead center based on
the developed velocity and acceleration observers. From Fig. 2 we can see that at π‘π΅π·πΆ the chamber
pressure is equal to the suction pressure, which from (1) shows that πΉπ(π‘π΅π·πΆ) = 0. Likewise, since bottom
dead center is by definition a local maximum of the position it can be seen that οΏ½ΜοΏ½(π‘π΅π·πΆ) = 0 as well, and
assuming convergence of the velocity observer that οΏ½ΜΜοΏ½π(οΏ½ΜοΏ½π΅π·πΆ) = 0 as well. Using this information the
mechanical dynamics of the system (2) and (3) at this instant of time can be simplified as
54
πΌπΌ(π‘π΅π·πΆ) = ποΏ½ΜοΏ½(π‘π΅π·πΆ) + πΎ(π₯π΅π·πΆ β πΏ0). (106)
Rewriting (106) and substituting the observed acceleration from the defined in (21) and the estimated
bottom dead center timing οΏ½ΜοΏ½π΅π·πΆ a bottom dead center observation can be defined as
οΏ½ΜοΏ½π΅π·πΆ β1
πΎ(πΌπΌ(οΏ½ΜοΏ½π΅π·πΆ) β ποΏ½ΜΜοΏ½π(οΏ½ΜοΏ½π΅π·πΆ)) + πΏ0. (107)
Note that a new value of οΏ½ΜοΏ½π΅π·πΆ can be calculated whenever π‘ = π‘π΅π·πΆ , i.e. once per cycle.
7.4 TOP DEAD CENTER OBSERVATION
An observation for top dead center can be obtained by rewriting (4) and substituting the stroke length and
bottom dead center observations defined in (105) and (107), respectively, to obtain the following
οΏ½ΜοΏ½ππ·πΆ β οΏ½ΜοΏ½π΅π·πΆ β βοΏ½ΜοΏ½ππΏ . (108)
55
8 REGULATION CONTROLLERS
Three different levels of control are proposed to achieve the system level control objectives of
resonance-tracking and stroke control. These controllers use the auxiliary observations derived in Section 7
to achieve this control sensorlessly. Collectively, these three controllers determine the form of the control
input to the system π£π(π‘). For the sake of simplicity this voltage command is selected to be a single
frequency sinusoid with variable amplitude and frequency based on the following definition:
π£π(π‘) β π0(π‘) sin π0(π‘). (109)
where π0(π‘) β β is the amplitude of the waveform and the phase π0(π‘) β β of the waveform is defined as
π0(π‘) β β« 2ππ0(π‘)π‘
π‘0
. (110)
where π0(π‘) β β is the frequency of the waveform. The form of π0(π‘) used in (110) is chosen to ensure
that the phase is piecewise continuous for step changes in π0(π‘).
As can be seen from (109) and (110) the indirect control variables which remain to be defined by the
subsequent controllers are the amplitude π0(π‘) and frequency, π0(π‘) of the voltage command π£π(π‘). A
diagram of the system level control scheme and its interaction with the observers developed previously is
shown in Fig. 18.
Figure 27. Block diagram of system level sensorless control scheme.
56
8.1 CURRENT CONTROLLER
The purpose of this controller is to regulate the magnitude of the force being applied to the piston by
controlling the motor current per the relationship described in (3). Rather than attempt to achieve full
tracking control of the motor current signal, a simpler case is considered in which the peak value of the
current is controlled to a desired level. The peak value of the motor current over the previous cycle,
denoted as πΌπ(π‘) β β, is here defined as
πΌπ(π‘) β max{|πΌ(π)| βΆ π β (π‘ β π, π‘]}. (111)
For a given desired peak current πΌπβ(π‘) β β a peak current error can be defined as
ππΌ(π‘) β πΌπβ(π‘) β πΌπ(π‘). (112)
The value of the voltage amplitude ππ(π‘) can be used as a control input for the current controller. A
proportional-integral form is chosen for this controller for simplicity and stability and can be defined as
π0(π‘) = ππππΌ(π‘) + ππΌβ« ππΌ(π)πππ‘
π‘0
. (113)
8.2 RESONANCE CONTROLLER
The purpose of the resonance controller is to manage the fundamental frequency of the system such
that resonance is achieved, thereby maximizing system efficiency. Based on the form of the voltage
command defined in (109) it can be shown that the resulting fundamental frequency of all system states
will be equal to π0.
The relative phase of the motor current and piston velocity is used as an indicator of resonance. In
this case, the observation of this phase as defined in (104) is used in place of the actual phase. An ideal
resonant phase is used as a reference for the controller and is denoted as πππΌβ β β. From this a phase error
can be defined as
ππ(π‘) β πππΌβ β οΏ½ΜοΏ½ππΌ . (114)
In the case of single-frequency sinusoidal current and velocity signals this desired phase would be zero to
indicate that motor force and velocity were completely in phase. However, since in reality current and
velocity will contain multiple harmonics, the phase as defined in (30) will likely be non-zero at resonance.
57
A search algorithm then utilizes this error signal to make adjustments to the excitation frequency. A
piecewise function is used to determine the size of the adjustment denoted as βπ β β based on the
magnitude and sign of the error as shown in (below).
βπ(ππ) β
{
βπ1π ππ(ππ) |ππ| > ππ1βπ2π ππ(ππ) ππ2 < |ππ| β€ ππ1βπ3π ππ(ππ) ππ3 < |ππ| β€ ππ2
0 |ππ| β€ ππ3
. (115)
where βπ1, βπ2, βπ3 β β+ are real positive constants such that βπ1 > βπ2 > βπ3, and ππ1 , ππ2 , ππ3 β β
+ are
real positive error threshold constants such that ππ1 > ππ2 > ππ3. Note that when the error is within the
smallest threshold the search algorithm no longer makes adjustments to the frequency. This condition is
included to prevent unnecessary hunting.
The rate at which these adjustments are made is determined by two conditions. The first condition is
that at least ππ seconds have passed since the most recent adjustment of this controller, where ππ β β is a
pre-determined period of time which is sufficiently long to allow the system dynamics to settle out after an
adjustment has been made. The second condition is that the controller has been given priority. The rule set
determining this priority is described in a subsequent section.
8.3 TOP DEAD CENTER CONTROLLER
The purpose of the top dead center controller is to control the distance between the piston and the
cylinder head. Given a desired top dead center value π₯ππ·πΆβ β β a top dead center error signal can be
defined as
ππ₯(π‘) β π₯ππ·πΆβ β οΏ½ΜοΏ½ππ·πΆ . (116)
where οΏ½ΜοΏ½ππ·πΆ could interchangeably be the observation obtained in either Sections 6.5 or 7.4. From this error
signal a search algorithm of the same structure as the one described in the previous subsection can then be
utilized to make adjustments to the desired peak current. This adjustment, denoted as βπΌπβ β β, is added to
the current value of πΌπβ(π‘) and is defined by the following piecewise function:
58
βπΌπβ(ππ₯) β
{
βπΌ1π ππ(ππ₯) |ππ₯| > ππ₯1βπΌ2π ππ(ππ₯) ππ₯2 < |ππ₯| β€ ππ₯1βπΌ3π ππ(ππ) ππ₯3 < |ππ₯| β€ ππ₯2
0 |ππ₯| β€ ππ₯3
. (117)
where βπΌ1, βπΌ2, βπΌ3 β β+ are real positive constants such that βπΌ1 > βπΌ2 > βπΌ3, and ππ₯1 , ππ₯2 , ππ₯3 β β
+ are
real positive error threshold constants such that ππ₯1 > ππ₯2 > ππ₯3.
The rate at which these adjustments are made is determined by two conditions, similar to the
resonance controller. The first condition is that at least ππ₯ seconds have passed since the most recent
adjustment of this controller, where ππ₯ β β is a pre-determined period of time which is sufficiently long to
allow the system dynamics to settle out after an adjustment has been made. The second condition is that
the controller has been given priority. The rule set determining this priority is described in the following
section.
8.4 CONTROLLER PRIORITY
Since all three of these controllers are to operate in parallel and the inputs and outputs of each are tied
via the system dynamics, special care needs to be taken to ensure stability of the system. In the case of the
top dead center controller the situation is even more dire, since its output is a reference to the current
controller, making it obvious that care must be taken to restrict the updating of this controller. For this
purpose a rule set has been developed to set the priority of these controllers. The goal of this prioritization
is to prevent multiple controllers from making large adjustments to their outputs at the same time, which
may cause them to fight each other and create instability.
Since the output adjustment of each of the three controllers presented increases with error, we can use
their error signals as an indication of how hard the respective controller is working. To facilitate this
prioritization, error bounds are established for the current and resonant controller errors defined in (112)
and (114), respectively. These error bounds and corresponding Boolean indicators are defined as
π½πΌ β {πππ’π |ππΌ| < ππΌ1πΉπππ π |ππΌ| β₯ ππΌ1
(118)
59
π½π β {
πππ’π |ππ| < ππ3πΉπππ π |ππ| β₯ ππ3
(119)
where ππΌ1 β β+ is a real positive error threshold constant.
The order of priority established for this system in descending order is current controller, resonant
controller, and then top dead center controller. Since the current controller is given highest priority, no
restrictions are placed upon its updating, i.e. it updates continuously. As the next highest priority
controller, the resonant controller is only allowed to make adjustments to its output when the current error
ππΌ is within its bound, i.e. when π½πΌ is βTrueβ. As the lowest priority controller, the top dead center
controller is only allowed to make adjustments to its output when both the current and resonant controller
errors are within their bounds, i.e. when π½πΌ β π½π is βTrueβ. When one of these conditions is false the
corresponding controller output is held at its current value until a βTrueβ value is detected.
8.5 EXPERIMENTAL RESULTS
The velocity and uncertainty observers defined in (55) and (58) were executed on the FPGA with the
parameters and gains listed in Table 3. The resulting οΏ½ΜΜοΏ½π(π‘) and οΏ½ΜΜοΏ½π(π‘) from these observers were then
utilized in real-time to obtain the auxiliary observations described in Section 7. These observations were
then utilized to achieve the three-level real-time control described in Section 8 using the parameters and
gains listed in Table 3. For this experiment the current controller was run with an initial desired peak
current value of πΌπβ(π‘) = 0.5 [π΄] and the resonance controller was run with an initial frequency of π0(π‘) =
56 [π»π§]. Based on experimental testing of the compressor efficiency an ideal phase between the positive
zero-crossing of the motor current and the piston velocity was identified as πππΌβ = 10Β°. A fixed desired top
dead center value of π₯ππ·πΆβ = 1 [ππ] was selected for this experiment.
60
Table 4
Observer and Controller Parameters for Sensorless Overall Control Scheme
Parameter Value Units Parameter Value Units
π 6.47 Ξ© βπ1 1 Hz
πΏ 0.318 H βπ2 0.1 Hz
πΌ 75.6 N/A βπ3 0.01 Hz
π 0.65 kg ππ1 20 Β°
πΎ 72800 N/m ππ2 5 Β°
π₯0 6.35 mm ππ3 1 Β°
π1π 24,000 βπΌ1 0.02 A
π2π 40,000 βπΌ2 0.01 A
π1π 4,000 βπΌ3 0.005 A
π2π 40,000 ππ₯1 2 mm
ππ 200 ππ₯2 0.5 mm
ππΌ 500 ππ₯3 0.1 mm
With these settings and the system level controller was turned on with the system at rest and allowed
to run until all three controllers had reached a steady-state at their respective desired values. The results of
the sensorless combined observer/controller hardware implementation are shown in Fig. 28-32. From Fig.
28 and 29 we can see that the current controller remains stable and convergent as the top dead center
controller increases the desired peak current. We can also see that the amplitude of the voltage command
π0(π‘) remains within the limits of the DC link voltage to prevent over-modulation. From the various
periods of time in Fig. 28 where the desired peak current remains constant we can see the effect of the
controller priority scheme. During these flat periods we can see that the top dead center controller is
waiting for the current and resonance controllers to reconverge before making further adjustments to πΌπβ(π‘).
In Fig. 30 and 31 we see that the observed phase οΏ½ΜοΏ½ππΌ(π‘) accurately observes the measured phase
πππΌ(π‘) and that the frequency of the voltage command π0(π‘) is successfully manipulated to achieve the
desired phase. We can see that each time the peak motor current is increased the phase is consistently
reduced giving the effect that phase error ππ(π‘) is more positive than negative. However, once steady-state
61
has been achieved from approximately 70 seconds onward, and current adjustments are minor if any, that
the phase error is zero-meaned about the desired phase.
In Fig. 32 we see that the observed top dead center οΏ½ΜοΏ½ππ·πΆ(π‘) accurately observes the measured top
dead center π₯ππ·πΆ(π‘) and that the peak current is successfully manipulated to achieve the desired top dead
center value. From all of these figures we can see that the rules of priority between the three controllers
successfully maintain system-wide stability.
Figure 28. Performance of proportional-integral peak current controller with varying setpoint determined
by top dead center controller.
Figure 29. Amplitude π0(π‘) of the voltage excitation π£π(π‘) calculated by the current controller.
62
Figure 30. Measured and observed phase difference between the piston velocity and measured current with
corresponding desired phase.
Figure 31. Frequency π0(π‘) of the voltage excitation set by the resonance controller as it attempts to
achieve the desired phase shown in Fig. 21.
Figure 32. Measured and observed top dead center with corresponding desired top dead center value.
63
9 TRACKING CONTROLLERS
9.1 CURRENT CONTROLLER
As a more sophisticated alternative to the regulation current controller described in Section 8.1 a
nonlinear current controller is to be developed for the linear motor to allow for trajectory tracking of the
motor current rather than controlling to a single point in the waveform as is currently done in Section 8.1.
Expert design of this current trajectory could allow for further increase in the system efficiency.
9.1.1 CONTROL DEVELOPMENT
For a given current trajectory πΌπ(π‘) β β, the designed controller should ensure that πΌ(π‘) β πΌπ(π‘) as
π‘ β β.The control input for the system is the motor voltage signal π£π(π‘). To facilitate the development of
the current controller the dynamic equation in (3) can be rewritten as
πΌΜ =1
πΏπ(π£π β π ππΌ) β ππΌ (120)
where ππΌ(π‘) β β denotes the uncertain scaled back EMF term πΌ
πΏποΏ½ΜοΏ½.
The proceeding mathematical analysis necessitates the following set of assumptions (previous
assumptions are ignored):
Assumption 1: The signal π£π(π‘) is assumed to be known and πΌ(π‘) is assumed to be measurable.
Assumption 2: The signals π₯(π‘), οΏ½ΜοΏ½(π‘), πΜ(π‘), πΜ(π‘) are all assumed to be piecewise continuous and
bounded, hence there exist positive bounding constants ν1, ν2 β β+ such that |πΜ(π‘)| < ν1, |πΜ(π‘)| < ν2
Assumption 3: The machine parameters πΏπ , π π , and πΌ are known a priori and are assumed to be
constants with respect to time.
Assumption 4: The current trajectory πΌπ(π‘) is chosen such that πΌπ(π‘) and πΌοΏ½ΜοΏ½(π‘) are piecewise
continuous and bounded. Through proper design of this trajectory we can ensure tracking of the system
resonant frequency.
64
The following error signals can be defined for the system:
ππΌ β πΌπ β πΌ (121)
οΏ½ΜοΏ½πΌ = πΌοΏ½ΜοΏ½ β πΌ.Μ (122)
Substituting (120) into (122) gives the following open loop error equation:
οΏ½ΜοΏ½πΌ = πΌοΏ½ΜοΏ½ β1
πΏπ(π£π β π ππΌ) β ππΌ . (123)
The form of (123) and the subsequent stability analysis motivate the following design for the control input:
π£π = πΏπ(πΌοΏ½ΜοΏ½ β ππΌ) + π ππΌ (124)
where π(π‘) β β is a subsequently designed observer of the back EMF term π(π‘). An error signal for this
observer can be defined as follows:
ππΌ = ππΌ β ππΌ . (125)
Substituting (124) into (123) and utilizing (125) gives the following closed loop error equation for the
system:
οΏ½ΜοΏ½πΌ = ππΌ . (126)
To facilitate the development of the back EMF observer a filtered error signal is defined as follows
π πΌ β οΏ½ΜοΏ½πΌ + ππΌ . (127)
From the subsequent stability analysis the following observer is designed:
πΜπΌ β (π1πΌ + 1)π πΌ + π2πΌπ ππ(ππΌ) (128)
where π1πΌ , π2πΌ β π + are constant gains. As stated in Assumption 1 the signal πΌ(Μπ‘) is not measurable,
meaning that οΏ½ΜοΏ½πΌ(π‘) and therefore π πΌ(π‘) are likewise unavailable. A realizable form of the observer can be
found by substituting (127) into (128) and integrating both sides of the resulting equation to obtain the
following:
ππΌ = (π1πΌ + 1) [ππΌ(π‘) β ππΌ(π‘0) + β«ππΌ(π)ππ
π‘
π‘0
] + π2πΌ β«π ππ(ππΌ(π))ππ
π‘
π‘0
+ ππΌ(π‘0). (129)
Remark 9.1.1: The stability analysis in the subsequent section proves that the motor current πΌ(π‘)
converges to the current trajectory πΌπ(π‘) or in other words that π(π‘) converges to zero. This stability
analysis will also show that οΏ½ΜοΏ½πΌ(π‘) converges to zero, which per (126) proves that ππΌ(π‘) converges to ππΌ(π‘).
From the definition of ππΌ(π‘) we can define an observed velocity as
65
οΏ½ΜΜοΏ½πΌ βπΏππΌππΌ . (130)
9.1.2 STABILITY ANALYSIS
Theorem 9.1.1: The observer defined in (129) ensures that
ππΌ(π‘) β ππΌ(π‘) ππ π‘ β β (131)
provided that the observer gain π2πΌ is selected to meet the following sufficient condition:
π2πΌ > ν1πΌ + ν2πΌ . (132)
Proof: See Appendix.
9.1.3 SIMULATION RESULTS
The simulation model detailed in Section 3 was utilized for this simulation with a switching frequency
of 25 [kHz] used for the inverter model. A single frequency sinusoidal current trajectory was chosen for
this simulation of the form πΌπ(π‘) = πΌ0 sin 2ππ0π‘ β (1 β πβπ½π‘3) where π0 = 63 [π»π§], π½ = 75,000 and πΌ0 is
initially 1.4 [π΄] then undergoes a step change to 2.0 [π΄] at time π‘ = 0.119 [π ππ]. It can be shown that this
form fulfills the requirements of Assumption 4. The controller gain π2πΌ = 400,000 was selected per the
requirements stated in (16) and a value of π1πΌ = 7,500 was chosen for the integral gain.
The results of this simulation are shown in Fig. 33-36. Fig. 33 shows the current trajectory πΌπ(π‘) along
with the actual current πΌ(π‘). From this and the error signal π(π‘) in Fig. 34 we can see that the controller is
successful in achieving the current control objective and that πΌ(π‘) β πΌπ(π‘) as π‘ β β. The voltage
command signal generated by the controller is shown in Fig. 35 along with the corresponding inverter
output voltage. Fig. 36 shows an observed velocity signal οΏ½ΜΜοΏ½πΌ(π‘) derived from the observed back EMF term
ππΌ(π‘) as shown in (14) versus the actual velocity οΏ½ΜοΏ½(π‘). Since it has been proven through the stability
analysis that ππΌ(π‘) β ππΌ(π‘) ππ π‘ β β, it can be shown that likewise that οΏ½ΜΜοΏ½πΌ(π‘) converges to οΏ½ΜοΏ½(π‘). This
statement is further validated by the results shown in Fig. 36.
66
Figure 33. Convergence of actual current πΌ(π‘) to the current trajectory πΌπ(π‘).
Figure 34. Convergence of current error ππΌ(π‘) to zero.
67
Figure 35. Controller voltage command π£π(π‘) and corresponding H-bridge inverter output.
Figure 36. Comparison of observed velocity οΏ½ΜΜοΏ½πΌ(π‘) with actual velocity οΏ½ΜοΏ½(π‘).
68
CONCLUSION
A series of estimators, observers, and control schemes have been developed to allow for a sensorless
system level control scheme which is capable of achieving efficient stroke control of a linear vapor
compressor. The system level control has been implemented using three individual controllers operating in
parallel with a simple rule set preventing fighting between them. The information needed for the operation
of these controllers has been derived sensorlessly from a pair of Lyapunov stable nonlinear observers. The
first of these is a velocity observer, which is used to do determine the relative phase of the motor current
and piston velocity, an indicator of system efficiency. The second is a position and pressure observer
which uses this velocity signal to observe the absolute position of the piston along with the system
pressures. Both of these observers rely on accurate knowledge of the machine parameters which have been
obtained through a pair of adaptive least-squares estimators. Analysis and experimental results have been
provided for validation of each of the proposed algorithms, demonstrating successful and accurate
operation.
69
REFERENCES
[1] R. L. Unger, "Linear compressors for non-CFC refrigeration," in Int. Appl. Tech. Conf., West
Lafayette, IN, May 1996, pp. 373-380.
[2] R. L. Unger, "Linear compressors for clean and specialty gases," in Int. Compress. Eng. Conf., West
LaFayette, IN, Jan. 1998, pp. 51-56.
[3] R. W. Redlich, R. Unger and N. van der Walt, "Linear compressors: motor configuration, modulation
and systems," in Int. Compress. Eng. Conf., West Lafayette, IN, July 1996, pp. 341-346.
[4] Y.-P. Yang and W.-T. Chen, "Dual stroke and phase control and system identification of linear
compressor of a split-Stirling cryocooler," Asian J. of Control, vol. 1, no. 2, pp. 116-121, June 1999.
[5] B. J. Huang and Y. C. Chen, "System dynamics and control of a linear compressor for stroke and
frequency adjustment," J. of Dyn. Syst., Meas., and Control, vol. 124, no. 1, pp. 176-182, Feb. 2001.
[6] R. W. Redlich, "Method and apparatus for measuring piston position in a free piston compressor".
U.S. Patent 5342176, 30 August 1994.
[7] J. Zhang, Y. Chang and Z. Xing, "Study on self-sensor of linear moving magnet compressor's piston
stroke," IEEE Sensors J., vol. 9, no. 2, pp. 154-158, Feb. 2009.
[8] J. Sung, C. Lee, G. Kim, T. Lipo, C. Won and S. Choi, "Sensorless control for linear compressors,"
Int. J. of Appl. Electromagn. and Mech., vol. 24, no. 3-4, pp. 273-286, Mar. 2006.
[9] T.-W. Chun, J.-R. Ahn, J.-Y. You and C.-W. Lee, "Analysis and control for linear compressor system
driven by PWM inverter," in 30th Annu. Conf. IEEE Ind. Electron. Soc., Busan, Korea, pp. 263-267,
Nov. 2004.
[10] J.-K. Son, T.-W. Chun, H.-H. Lee, H.-G. Kim and E.-C. Nho, "Method of estimating precise piston
stroke of linear compressor driven by PWM inverter," in 16th Int. Power Electron. and Motion
Control Conf. and Expo., Antalya, Turkey, Sept. 2014, pp. 673-678.
[11] Z. Lin, J. Wang and D. Howe, "A resonant frequency tracking technique for linear vapor
compressors," in Elect. Mach. & Drives Conf., Antalya, Turkey, May 2007, pp. 370-375.
[12] T.-W. Chun, J.-R. Ahn, H.-H. Lee, H.-G. Kim and E.-C. Nho, "A novel strategy of efficiency control
for a linear compressor system driven by a PWM converter," IEEE Trans. Ind. Electron., vol. 55, no.
1, pp. 296-301, Jan. 2008.
[13] S.-B. Yang, "Linear compressor control circuit to control frequency based on the piston position of the
linear compressor". United States of America Patent 5947693, 7 September 1999.
[14] M. Yoshida, S. Hasegawa and M. Ueda, "Driving apparatus of a linear compressor". United States of
America Patent 6832898, 21 December 2004.
[15] B.-J. Choi and e. al, "Linear compressor". United States of America Patent 7,816,873, 9 March 2009.
[16] Z. Lin, J. Wang and D. Howe, "A learning feed-forward current controller for linear reciprocating
vapor compressors," IEEE Trans. Ind. Electron., vol. 58, no. 8, pp. 3383-3390, Aug. 2011.
[17] Y. Zhu and P. Pagilla, "Adaptive Estimation of Time-Varying Parameters in Linear Systems," in Proc.
Amer. Control Conf., Denver, CO, 2003, pp. 4167-4172.
[18] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness, Englewood
Cliffs, NJ: Prentice Hall Inc., 1989.
[19] M. McIntyre, T. Burg, D. Dawson and B. Xian, "Adaptive state of charge (SOC) estimator for a
battery," in Proc. Amer. Control Conf., Minneapolis, MN, 2006, pp. 5740-5744.
[20] M. Salah, M. McIntyre, D. Dawson and J. Wagner, "Sensing of the time-varying angular rate for
MEMS Z-axis gyrsocopes," IEEE Trans. Mechatron., vol. 20, no. 6, pp. 720-727, 2010.
70
[21] E. Tatlicioglu, M. McIntyre, D. Dawson and I. Walker, "Adaptive Nonlinear Tracking Control of
Kinematically Redundant Robot Manipulators with Sub-Task Extensions," in 44th IEEE Conf. on
Decision and Control, Seville, 2005.
[22] D. W. Hart, "Inverters," in Power Electronics, New York, McGraw-Hill, 2011, ch. 8, sec. 10, pp. 358-
359.
[23] J. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall, 1991.
[24] G.-S. Choe and K.-J. Kim, "Analysis of nonlinear dynamics in a linear compressor," JSME Int. J., vol.
43, no. 3, pp. 545-552, Aug. 2000.
[25] J. Latham, M. L. McIntyre and M. Mohebbi, "Parameter estimation and a series of nonlinear observers
for the system dynamics of a linear vapor compressor," IEEE Trans. Ind. Electron., vol. 63, no. 11, pp.
6736-6744, Nov. 2016.
71
APPENDIX
Remark A.1: We can see that a similar RISE based observer methodology was used to develop velocity,
acceleration, and gas force observers in Sections VI.A, VI.B, VI.D respectively as well as the observer-
based current controller in Section IX.A. Given that the closed loop error dynamics and the form of the
observers defined in (33) and (47) are identical a single stability analysis is given for all three systems for
the sake of brevity. The subscript π used in the subsequent analysis may be replaced with a π,π, πΉ, or πΌ to
refer to terms in the velocity observer, acceleration observer, gas force observer, or current controller
development, respectively.
The relevant error definitions, closed loop dynamics, and observer definitions are repeated here for use in
the subsequent analysis. Corresponding equations can be found in the respective development sections.
ππ β ππ β ππ (133)
οΏ½ΜοΏ½π = ππ (134)
π π β οΏ½ΜοΏ½π + ππ (135)
πΜπ β (π1π + 1)π π + π2ππ ππ(ππ) (136)
Corresponding equations can be found in the development sections mentioned.
Theorem A1: The observer defined in (136) ensures that
ππ(π‘) β ππ(π‘) ππ π‘ β β (137)
provided that the observer gain π2π is selected to meet the following sufficient condition:
π2π > ν1π + ν2π . (138)
Proof: To prove that ππ(π‘) β ππ(π‘) as π‘ β β, a nonnegative Lyapunov function ππ(π‘) β β is defined for
the observer as
ππ β1
2ππ2 +
1
2π π2 (139)
72
where ππ(π‘) is the error of the respective controller/observer and π π(π‘) is the corresponding filtered error
defined in (135). Further analysis will show that the form of πΜπ(π‘) presented in (136) ensures that οΏ½ΜοΏ½π(π‘) is
negative definite for all ππ(π‘) and π π(π‘). The time derivative of (139) is taken and can be written as
οΏ½ΜοΏ½π = πποΏ½ΜοΏ½π + π π οΏ½ΜοΏ½π . (140)
An expression for οΏ½ΜοΏ½π(π‘) can be written by taking the time derivative of (135), and substituting the time
derivatives of (133) and (134) to obtain the following:
οΏ½ΜοΏ½π = ποΏ½ΜοΏ½ β πΜπ + οΏ½ΜοΏ½π . (141)
Solving (135) for οΏ½ΜοΏ½π(π‘) an expression is obtained which can be substituted along with (141) into (140) with
the following result:
οΏ½ΜοΏ½π = βππ2 + π π
2 + π π(ποΏ½ΜοΏ½ β ππΜ ). (142)
Substituting (135) and (136) into (142), after simplifying the result can be written as
οΏ½ΜοΏ½π(π‘) = βππ2 β π1ππ π
2 + οΏ½ΜοΏ½πποΏ½ΜοΏ½ β π2ποΏ½ΜοΏ½ππ ππ(ππ) + ππ(ποΏ½ΜοΏ½ β π2ππ ππ(ππ)). (143)
The integral of (143) from π‘0 to π‘ can be expressed as
ππ(π‘) = ππ(π‘0)ββ« ππ2(π)ππ
π‘
π‘0
β π1πβ« π π2(π)ππ
π‘
π‘0
+β« οΏ½ΜοΏ½π(π)ποΏ½ΜοΏ½(π)πππ‘
π‘0
βπ2πβ« οΏ½ΜοΏ½π(π)π ππ(ππ(π))πππ‘
π‘0
+ β« ππ(π)[ποΏ½ΜοΏ½(π)β π2ππ ππ(ππ(π))]πππ‘
π‘0
.
(144)
After integrating the fourth term of the right-hand side of (144) by parts and the fifth term with respect to
time and rearranging terms the following expression is obtained for ππ(π‘):
ππ(π‘) = ππ(π‘0) β β« ππ2(π)ππ
π‘
π‘0
β π1πβ« π π2(π)ππ
π‘
π‘0
+ ππ(π‘)ποΏ½ΜοΏ½(π‘) β ππ(π‘0)ποΏ½ΜοΏ½(π‘0)
(145)
βπ2π|ππ(π‘)| + π2π|ππ(π‘0)| + β« ππ(π)[ποΏ½ΜοΏ½(π)β ποΏ½ΜοΏ½(π)β π2ππ ππ(ππ(π))]πππ‘
π‘0
.
Provided that π2π is selected according to (138), the fourth and sixth terms of the right-hand side of (145)
can be combined and upper bounded to zero. Similarly, the eighth term can also be upper bounded to zero.
After applying these upper bounds to (145), ππ(π‘) can be upper bounded as follows:
ππ(π‘) β€ββ« |ππ(π)|2ππ
π‘
π‘π
β π1πβ« |π π(π)|2ππ
π‘
π‘π
+ πΆπ. (146)
where πΆπ β ππ(π‘0) β ππ(π‘0)(ποΏ½ΜοΏ½(π‘0) β π2ππ ππ(ππ(π‘0))) is a bounding constant.
73
From the structure of (139) and (146) and the definition of πΆπ it is proven that ππ(π‘) β ββ; hence
π π(π‘), ππ(π‘) β ββ. Since π π(π‘), ππ(π‘) β ββ, from (135) and (136) it is clear that οΏ½ΜοΏ½π(π‘), πΜπ(π‘) β ββ. Since
ποΏ½ΜοΏ½(π‘), πΜπ(π‘), π π(π‘), ππ(π‘) and οΏ½ΜοΏ½π(π‘) β ββ, it is clear from (141) that οΏ½ΜοΏ½π(π‘) β ββ. The inequality defined by
(146) can be used to prove that π π(π‘), ππ(π‘) β β2. Since π π(π‘), ππ(π‘), οΏ½ΜοΏ½π(π‘), οΏ½ΜοΏ½π(π‘) β ββ and π π(π‘), ππ(π‘) β
β2, then Barbalatβs lemma can be used to prove that π π(π‘) and ππ(π‘) β 0 as π‘ β β. Since π π(π‘) β 0 as π‘ β
β, (135) can be used to show that οΏ½ΜοΏ½π(π‘) β 0 as π‘ β β as well. From this fact and (134) we see that ππ(π‘) β
ππ(π‘) thus completing the proof of Theorem A1.
74
CURRICULUM VITAE
NAME: Joseph Wilson Latham
ADDRESS: Department of Electrical and Computer Engineering
W. S. Speed Hall, Rm. 207
University of Louisville
Louisville, KY 40292
DOB: Greenville, Kentucky β August 19, 1988
EDUCATION
& TRAINING: B.S., Electrical Engineering
Western Kentucky University
2006-2012
M.S., Electrical and Computer Engineering
University of Louisville
2013-2014
Ph.D., Electrical Engineering
University of Louisville
2014-2017
AWARDS: Outstanding Graduate Student
Dept. of Electrical and Computer Engineering
University of Louisville
2016-2017
University Fellowship
University of Louisville
2014-2016
Award of Excellence Scholarship
Western Kentucky University
2006-2007, 2010-2012
PUBLICATIONS:
J. Latham, M. L. McIntyre and M. Mohebbi, "Sensorless resonance-tracking and stroke control of a
linear vapor compressor via nonlinear observers," IEEE Trans. Ind. Electron., under review, 2016.
J. Latham, M. L. McIntyre and M. Mohebbi, "Parameter estimation and a series of nonlinear
75
observers for the system dynamics of a linear vapor compressor," IEEE Trans. Ind. Electron., vol. 63, no.
11, pp. 6736-6744, Nov. 2016.
M. L. McIntyre, W. Burke, J. Latham and J. Graham, "Smart appliances energy study," Int. J. of
Comput. and their Applicat., vol. 21, no. 3, pp. 142-150, 2014.
J. Latham, M. L. McIntyre and M. Mohebbi, "Nonlinear adaptive current control for linear vapor
compressors," in 2016 IEEE Int. Conf. on Automat. Sci. and Eng. (CASE), Fort Worth, TX, pp. 1157-1162,
Aug. 2016.
M. L. McIntyre, M. Mohebbi and J. Latham, "Nonlinear current observer for backstepping control of
buck-type converters," in IEEE 16th Workshop on Control and Modeling for Power Electron. (COMPEL),
Vancouver, pp. 1-7, July 2015.
M. Mohebbi, M. L. McIntyre and J. Latham, "Energy efficient DC to AC power conversion using
advanced controllers and novel voltage trajectories," in IEEE 16th Workshop on Control and Modeling for
Power Electron. (COMPEL), Vancouver, pp. 1-8, July 2015.
M. Mohebbi, M. L. McIntyre and J. Latham, "Vehicle to grid utilizing a backstepping controller for
bidirectional full-bridge converter and five-level active neutral point inverter," in IEEE 16th Workshop on
Control and Modeling for Power Electron. (COMPEL), Vancouver, pp. 1-7, July 2015.
M. Elliot, M. L. McIntyre and J. Latham, "Partial shading abatement through cell level inverter
system topology," in IEEE 42nd Photovoltaic Specialist Conf. (PVSC), New Orleans, pp. 1-5, June 2015.
M. L. McIntyre, W. Burke, J. Latham and J. Graham, "Energy usage patterns in fielded smart
appliances," in 29th Int. Conf. on Comput. and their Applicat. (CATA), Las Vegas, pp. 181-188, March
2014.
M. L. McIntyre, M. Schoen and L. J., "Simplified adaptive backstepping control of Buck DC:DC
converter with unknown load," in IEEE 14th Workshop on Control and Modeling for Power Electron.
(COMPEL), Salt Lake City, pp. 1-7, June 2013.
M. L. McIntyre, M. Schoen and J. Latham, "Backstepping control of a capacitance-less photovoltaic
power converter with maximum power point tracking," in IEEE 39th Photovoltaic Specialist Conf. (PVSC),
76
Tampa, pp. 2908-2913, June 2013.
PATENTS:
J. W. Latham, M. L. McIntyre, G. Hahn and S. Kusumba, "A method for operating a linear
compressor". U.S. Patent Docket Number 280752-1/GECA-1135, filed Nov. 2015, (unique claims for
system level control scheme).
J. W. Latham and M. L. McIntyre, "A method for operating a linear compressor". U.S. Patent App.
#14/607,365, filed Jan. 2015, (unique claims for top dead center observation).
J. W. Latham and M. L. McIntyre, "A method for operating a linear compressor". U.S. Patent App.
#14/607,347, filed Jan. 2015, (unique claims for stroke length observation).
J. W. Latham and M. L. McIntyre, "A method for operating a linear compressor". U.S. Patent App.
#14/607,374, filed Jan. 2015, (unique claims for parameter estimation).
INVITED PRESENTATIONS:
"A backstepping controller for voltage source inverter with inductor current and output current
observers," in Power and Energy Conf. at Illinois (PECI), Champaign, 2017.
"Output feedback control of a single phase voltage source inverter utilizing a variable structure
observer," in American Controls Conf., Seattle, 2017.